Integral Transforms and Deformations of K3 Surfaces

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Jul 11, 2015 - O'Grady, and Yoshioka, which show that there is an isomorphism of Hodge ..... grateful to Paul Balmer and Matthias Künzer, and especially to ...
INTEGRAL TRANSFORMS AND DEFORMATIONS OF K3 SURFACES

arXiv:1507.03108v1 [math.AG] 11 Jul 2015

E. MARKMAN AND S. MEHROTRA Abstract. Let X be a K3 surface and M a smooth and projective moduli space of stable sheaves on X of Mukai vector v. A universal sheaf U over X × M induces an integral transform ΦU : Db (X) → Db (M ) from the derived category of coherent sheaves on X to that on M . (1) We prove that ΦU is faithful. ΦU is not full if the dimension of M is ≥ 4. (2) We exhibit the full subcategory of Db (M ), consisting of objects in the image of ΦU , as the quotient of a category, explicitly constructed from Db (X), by a natural congruence relation defined in terms of the Mukai vector v. (3) Let M0 be a component of the moduli space of isomorphism classes of marked irreducible holomorphic symplectic manifolds deformation equivalent to the Hilbert scheme X [n] of n points on a K3 surface X, n ≥ 2. M0 is 21-dimensional, while the moduli of K¨ ahler K3 surfaces is 20-dimensional. We construct a geometric deformation of the derived categories of K3 surfaces over a Zariski dense open subset of M0 , which coincides with Db (X) whenever the marked manifold is a moduli space of sheaves on X satisfying a technical condition. Statement (3) assumes Conjecture 1.6 asserting that the dimension, of the first sheaf cohomology of a reflexive hyperholomorphic sheaf with an isolated singularity, remains constant along twistor deformations. The conjecture is known to hold for hyperholomorphic vector bundles.

Contents 1. Introduction 1.1. A faithful functor 1.2. The full subcategory of D b (M, θ) with objects coming from the surface 1.3. A reconstruction of D b (X) as a category of comodules of a comonad 1.4. A deformation of K3-categories 1.5. Notational conventions 2. A universal monad in D b (X × X) 2.1. The monad associated to a morphism of varieties 2.2. A splitting of the monad 2.3. Splitting of the monad in the Hilbert scheme case 2.4. Hochschild (co)homology 2.5. Splitting of the monad for a general moduli space 3. Yoneda algebras 3.1. A congruence relation associated to the natural transformation h 3.2. The monad A is a quotient of a constant monad 3.3. A universal relation “ideal” 3.4. Computation of the full subcategory D b (X)T of D b (M ) 3.5. Traces 3.6. A relation in the Yoneda algebra of ΦU (x) 3.7. The natural transformation h2 is the Mukai vector Date: July 14, 2015. 1

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3.8. Moduli spaces of sheaves over the moduli space M 4. The transposition of the factors of M × M 5. A simple and rigid comonad in D b (M × M ) 5.1. F is simple and rigid 5.2. E is simple and rigid 6. A deformation of the derived category D b (X) 6.1. Deformability of F 6.2. The triangulated structure on the category of comodules 6.3. Monodromy invariance 6.4. A K3 category 7. Comparison with Toda’s Category 7.1. Hochschild cohomology and deformations 7.2. A map on tangent spaces 7.3. The comparison 8. Variations of Hodge structures References

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1. Introduction 1.1. A faithful functor. Let X be a projective K3 surface, H an ample line bundle on X, and M := MH (v) the (coarse) moduli space of Gieseker-Simpson H-stable sheaves E on √ X whose Chern classes are specified by the vector v = v(E ) := ch(E ) tdX in the integral cohomology of X. M is a smooth holomorphic-symplectic variety, by [Mu1]. We shall assume here that there aren’t any strict semi-stable sheaves of class v, so that M is projective. There is a numerical criterion which guarantees this: it amounts to choosing H to be v-generic, that is, in a region complementary to a countable union of suitable hyperplanes depending on v. While X × M need not carry a universal sheaf, there always exists a quasi-universal sheaf on it (see the appendix of [Mu3]). In other words, there is a twisted sheaf U on X × M which is universal locally on M in the etale or analytic topology (see [C1]). Write πX and πM for the two projections from X × M , and let ΦU : D b (X) → D b (M, θ) be the integral transform (1.1)

ΦU ( )

:=

∗ πM,∗ (πX ( ) ⊗ U ).

Here θ is the class of U in the Brauer group of M , D b (M, θ) stands for the bounded derived cat∗ are derived functors appropriately egory of θ-twisted coherent sheaves on M , and πM,∗ and πX defined in this context. We refer the reader to [C1] for a careful discussion of this formalism. Suffice it to say that most of the familiar properties of duality and the usual functorial calculus for derived categories of coherent sheaves continue to hold here. In particular, it follows by ∗ ( ) ⊗ U ∨ )[2]. Serre duality for twisted sheaves that ΦU has a right adjoint ΨU ( ) := πX,∗ (πM b b The composition ΨU ◦ ΦU : D (X) → D (X) is the integral transform with kernel (1.2)

∗ ∗ A := π13,∗ (π1∗ ωX ⊗ π12 (U )∨ ⊗ π23 (U ))[2] ∈ D b (X × X),

where πij is the projection from X × M × X onto the product of the i-th and j-th factors. ∗ e The Mukai lattice H(X, Z) of the K3 surface R ∨X is the integral cohomology H (X, Z) endowed with the Mukai pairing (u, v) := − S u ∪ v, where the duality operator u 7→ u∨ √ √ changes the sign of the direct summand in H 2 (X, Z). If u = ch(E) tdX and v = ch(F ) tdX are the Mukai vectors of two coherent sheaves E and F on X, then (u, v) = −χ(E ∗ ⊗ F ), by

INTEGRAL TRANSFORMS AND DEFORMATIONS

3

Hirzebruch-Riemann-Roch, where the dual E ∗ and the tensor product are taken in the derived category, and χ is the Euler characteristic. The dimension of the moduli space M := MH (v) discussed above is 2 + (v, v). Denote by ∆ : X → X × X the diagonal embedding. e Theorem 1.1. (Theorem 2.2). Let v ∈ H(X, Z) be a primitive class with (v, v) = 2n − 2, n ≥ 2, and H a v-generic polarization. One has a natural morphism α:

n−1 M i=0

∆∗ OX ⊗C Ext2i (OM , OM ) [−2i] → A .

(1) When M := MH (v) is the Hilbert scheme of n points on X and U is the universal ideal sheaf the morphism α is an isomorphism. In particular, a choice of a non-zero element of the one-dimensional vector space Ext2 (OM , OM ) yields an isomorphism n−1 A ∼ ∆∗ OX [−2i]. = ⊕i=0 (2) In general, for v arbitrary, the structure sheaf of the diagonal ∆∗ OX is a direct summand of A in D b (X×X). In particular, the integral transform ΦU : D b (X) → D b (M, θ) is faithful. Part (1) of Theorem 2.2 was independently proven by Nick Addington [Ad]. The isomorn−1 phism A ∼ ∆∗ OX [−2i] was established for some moduli spaces of torsion sheaves on = ⊕i=0 K3 surfaces in [ADM]. Further cases where α is an isomorphism are provided in Lemmas 6.9 and 6.16. The proof of the first part of Theorem 1.1 relies heavily on Haiman’s work [Ha1, Ha2] on the n!-Conjecture on one hand, and the derived McKay correspondence of Bridgeland-King-Reid on the other [BKR]. The second part makes use of results of Mukai, O’Grady, and Yoshioka, which show that there is an isomorphism of Hodge structures between H 2 (MH (v), Z) and the orthogonal complement v ⊥ of the Mukai vector v in the Mukai lattice of X ([Mu4, OG1, Y1], see also Theorem 2.16 below). 1.2. The full subcategory of D b (M, θ) with objects coming from the surface. Denote by D b (X)T the full subcategory of D b (M, θ) with objects of the form ΦU (x), for some object x in D b (X). We provide next an explicit computation of D b (X)T . We will use the following standard construction in category theory. Definition 1.2. [Mac, Section II.8]. Let C be a category. (1) A congruence relation R on C consists of an equivalence relation Rx1 ,x2 on Hom(x1 , x2 ), for every pair of objects x1 , x2 in C , satisfying the following property. Given morphisms f1 , f2 : x1 → x2 related by Rx1 ,x2 , objects x0 , x3 of C , and morphisms e : x0 → x1 and g : x2 → x3 , the morphisms gf1 e and gf2 e are related by Rx0 ,x3 in Hom(x0 , x3 ). (2) Let R be a congruence relation on C . The quotient category C /R is the category whose objects are those of C and such that HomC /R (x1 , x2 ) := HomC (x1 , x2 )/Rx1 ,x2 . The natural functor Q : C → C /R is called the quotient functor. An explicit computation of D b (X)T as a quotient category requires an explicit category C and an explicit relation R. We take C to be the full subcategory D b (X)Ye of D b (X × M ) ∗ (x), for some object x in D b (X). Set Hom• with objects of the form πX D b (X)e (x, y) :=

⊕i∈Z HomDb (X)e (x, y[i]). The category D b (X)Ye is explicit, since

Y

Y

Hom•Db (X)e (x, y) Y

∗ ∗ x, πX y) ∼ := Hom•Db (X×M ) (πX = Hom•Db (X) (x, y) ⊗ Hom•Db (M ) (OM , OM ).

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We describe next a congruence relation on D b (X)Ye . Set pt := Spec(C) and let c : M → pt be the constant morphism. We get the object Y (OM ) := Rc∗ OM in D b (pt). As a graded vector space Y (OM ) is ⊕ni=0 H 2i (M, OM )[−2i], where n = dimC (M )/2. Given a graded vector space V , let 1Db (X) ⊗C V be the endofunctor of D b (X) sending an object x to x ⊗C V . Set Υ := 1Db (X) ⊗C Y (OM ),

R := 1Db (X) ⊗C H 2n (M, OM )[−2n].

These two endofunctors are integral transforms with kernels Y := ∆∗ OX ⊗C Y (OM ) and R := ∆∗ OX ⊗C H 2n (M, OM )[−2n] in D b (X×X). Let πX : X×M → X be the projection. Note ∗ . We thus have the adjunction that the endo-functor Υ is naturally isomorphic to RπX∗ ◦ πX isomorphism (x, y) := Hom b (π ∗ x, π ∗ y) ∼ (1.3) Hom b = Hom b (x, Υ(y)). D (X)Y e

D (X×M )

X

X

D (X)

The congruence relation R is defined in terms of a natural transformation h : R → Υ, which we define Pn next. The natural transformation h is induced by a morphism h : R → Y . Write h = i=0 h2i according to the direct sum decomposition of Y (OM ), so that h2i : ∆∗ (OX ) ⊗C H 2n (M, OM )[−2n]



∆∗ (OX ) ⊗C H 2n−2i (M, OM )[2i − 2n].

Mukai’s Hodge isometry (2.18) provides a canonical identification of H 2 (X, OX ) with H 2 (M, OM ). ⊗i )[2i]), It follows that the morphism h2i is naturally a class in HomX×X (∆X,∗ (OX ), ∆X,∗ (ωX as we carefully check in Section 3. In particular, h2 is a class in the Hochschild homology HH0 (X). The Hochschild-Kostant-Rosenberg isomorphism (reviewed in Section 2.4) maps the Mukai vector v to a class in HH0 (X). We set h2 to be the image of v, h0 := −1, and h4 := −(h2 )2 . We necessarily have h2i = 0, for i > 2, for dimension reasons. Given an object x in D b (X), let hx : x ⊗C H 2n (M, OM )[−2n] → x ⊗C Y (OM )

be the morphism induced by the natural transformation h. Consider the relation R on D b (X)Ye given as follows. The morphisms f1 , f2 in HomDb (X)e (x1 , x2 ) are related by Rx1 ,x2 , if and only Y if f1 − f2 belongs to the image of the homomorphism (1.4)

(hx2 )∗ : HomDb (X) (x1 , x2 ) ⊗ H 2n (M, OM )[−2n] → HomDb (X)e (x1 , x2 ) Y

induced by composition with hx2 . It is easy to check that R is a congruence relation, as we do in the proof of Theorem 3.3. We describe next the functor inducing the equivalence between D b (X)T and the quotient category D b (X)Ye /R. Let ΞU : D b (X × M ) → D b (M, θ) be the composition of tensorization ∗ . Let Q : D b (X) → D b (X) be the restriction by U followed by RπM∗ . Then ΦU = ΞU ◦ πX T e Y of the functor ΞU . Theorem 1.3. (Theorem 3.3) Assume that the morphism α in Theorem 1.1 is an isomorphism. (1) R is a congurence relation. (2) The kernel of Q : HomDb (X)e (x, y) → HomDb (M,θ) (ΦU (x), ΦU (y)) is identified with Y the image of (hy )∗ : HomDb (X) (x, R(y)) → HomDb (X) (x, Υ(y)) via the adjunction isomorphism (1.3).

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(3) The functor Q is full. Consequently, Q factors as the composition of the quotient functor D b (X)Ye → D b (X)Ye /R and an equivalence functor D b (X)Ye /R ∼ = D b (X)T .

As a consequence we obtain the description of the Yoneda algebra of ΦU (F ), for a simple sheaf F on X, as the quotient of the tensor product of the Yoneda algebras of F and OM by the principal ideal generated by an explicit relation (Theorem 3.21). Related results were obtained by A. Krug for the Hilbert scheme of points on a projective surface [K]. Let us describe the key ingredient in the proof of parts (2) and (3) of Theorem 1.3. The ∗ gives rise to a natural transformation q : Υ → Ψ ◦ Φ equality ΦU = ΞU ◦ πX U U given by ≈ ≈ † ∗ q := RπX∗ η πX , where η : 1Db (X×M ) → ΞU ◦ ΞU is the unit for the adjunction ΞU ⊣ Ξ†U . The natural transformation q, in turn, is induced by a morphism of kernels q : Y → A . We get an exact triangle in D b (X × X), which admits a splitting q

h

(1.5)

0

R → Y → A → R[1],

by Proposition 3.9 and Theorem 3.10. The functor Q of Theorem 1.3 is induced by the natural transformation q : Υ → ΨU ◦ ΦU in the sense that the following diagram commutes q

HomDb (X) (x, Υ(y))

/ Hom b D (X) (x, ΨU ΦU (y)) ∼ =

∼ =





∗ x, π ∗ y) HomDb (X×M ) (πX X

ΞU

/ Hom b D (M,θ) (ΦU (x), ΦU (y)),

where the vertical arrows are the adjunction isomorphisms (Theorem 3.2). Parts (2) and (3) of Theorem 1.3 follow immediately from the splitting of the exact triangle (1.5). 1.3. A reconstruction of D b (X) as a category of comodules of a comonad. We shall be using the categorical device of (co)monads and (co)modules over them in order to recover the category D b (X) in terms of data in D b (M × M ). The latter data will then be deformed with M yielding non-commutative deformations of D b (X). Let us briefly recall the necessary notions here. A detailed presentation can be found in Chapter VI of Mac Lane’s text [Mac]. A comonad L on a category A is simply a comonoid object in the functor category End(A). Explicitly, L is a triple hL, ǫ, δi where L : A → A is an endofunctor, and the counit ǫ, and comultiplication δ are natural transformations ǫ : L −→ I, δ : L −→ L2 ,

satisfying coassociativity: (1.6)

L δ

δ

/ L2



L2





/ L3

δL

and the left and right counit laws: (1.7)

L

IL o

L 

ǫL

L

δ

L2



/ LI

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Associated to any adjunction F : X → A, G : A → X, with F ⊣ G, there is a natural comonad with L = F G : A → A, ǫ the counit of the adjunction, and δ = F ηG : L → L2 , where η is the unit of the adjunction. A comodule for a comonad hL, η, δi is a pair (a, h) consisting of an object a ∈ A and an arrow h : a → La such that δ ◦ h = Lh ◦ h and ǫ ◦ h = id. A morphism f : (a, h) → (a′ , h′ ) is an arrow f ∈ HomA (a, a′ ) which renders commutative the diagram a

h

/ La Lf

f





a′

h′

/ La′ .

The set of all L-comodules together with their morphisms form a category AL . Finally, monads, and modules over monads are simply the notions dual to those defined above. Adjoint pairs of functor naturally give rise to (co)monads. In our situation, let L be the composition ΦU ◦ ΨU : D b (MH (v), θ) → D b (MH (v), θ). The adjunction ΦU ⊣ ΨU yields the unit η : ΨU ◦ ΦU →id, the counit ǫ : ΦU ◦ ΨU →id, and the comultiplication (1.8)

δ = ΦU ηΨU : L→L2 .

These define a comonad L := hL, ǫ, δi in D b (MH (v), θ). We get the category D b (MH (v), θ)L of comodules of L. Denote by ΦU : D b (MH (v), θ)L → D b (MH (v), θ) the forgetful functor ΦU (G , h) = G . For (co)monads given by adjunctions, there is a natural comparison functor from the source category to the category of (co)modules, which in our case will be denoted as ˆ : D b (X)−→D b (MH (v), θ)L . It takes H ∈ D b (X) to the comodule (ΦU (H ), ΦU ηH ) Φ The unit η : id→ΨU ◦ ΦU is split by Theorem 1.1. Hence the triangulated version of the Barr-Beck Theorem [E, MS] (see also [Bal2]) immediately yields: ∼ = ˆ : D b (X) −→ D b (MH (v), θ)L is an equivalence of Proposition 1.4. The comparison functor Φ categories. In particular, it induces the structure of a triangulated category on D b (MH (v), θ)L such that the forgetful functor ΦU is exact.

This captures D b (X) in terms of structures defined only on MH (v). 1.4. A deformation of K3-categories. The deformations of MH (v) arising from those of the underlying K3 surface X form a 20-dimensional locus in the 21-dimensional Kuranishi deformation space of MH (v). Thus the generic deformation of MH (v) is not a moduli space of sheaves. The next theorem interprets these as arising from “non-commutative” perturbations of X. More precisely, we construct deformations of D b (X) which correspond to those of MH (v) away from the moduli space locus. Let F ∈ D b (MH (v) × MH (v), π1∗ θ −1 π2∗ θ) be the kernel of the endofunctor L. The object F plays a prominent role in the study of the geometry and cohomology of the moduli spaces MH (v) (see [Mu3, KLS, Ma1]). Let πij be the projection from MH (v) × X × MH (v) onto the product of the i-th and j-th factors. F is a complex with cohomology concentrated in degrees −1 and 0:  O∆M (v) if i = 0, ∗ ∗ H (1.9) H i (F ) = H omπ13 (π12 U , π23 U [i]) ∼ = E if i = −1, where E is a twisted reflexive sheaf of rank (dim MH (v) − 2).

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Definition 1.5. We shall refer to F as the modular complex of MH (v), and to E as the modular sheaf of MH (v) to emphasize their origin. The sheaf of Azumaya algebras E nd(E ) will be referred to as the modular Azumaya algebra of MH (v). Let X be a K3 surface with a trivial Picard group and let F ∈ D b (X [n] × X [n] ) be the kernel of the endofunctor L, so that its square L2 has kernel the convolution F ◦ F . By Proposition 5.1 of [C2], the counit ǫ and the comultiplication δ of L correspond to morphisms of objects F → O∆ and F → F ◦ F , respectively; denote these by ǫ and δ also. Thus we have the comonad object (1.10)

hF , ǫ, δi

in D b (X [n] × X [n] ) representing the comonad L. We recall here some of the results obtained in [MM2] which are required for the statement of the deformability of the comonad object given in Equation (1.10) (Theorem 1.8). A holomorphic symplectic compact K¨ ahler manifold M is said to be of K3[n] -type, if it is deformation equivalent to the Hilbert scheme X [n] of length n subschemes of a K3 surface X. The second integral cohomology of M is endowed with a non-degenerate integral symmetric bilinear pairing of signature (3, 20) called the Beauville-Bogomolov-Fujiki pairing [Be2]. Fix a lattice Λ isometric to H 2 (X [n] , Z). A marking for such M is an isometry η : H 2 (M, Z) → Λ. Let A be a reflexive sheaf of Azumaya algebras on M × M that is infinitesimally rigid, slope-stable with respect to some K¨ ahler class ω on M, and having the same numerical invariants as the modular Azumaya algebra E nd(E ) above (see [MM2, Section 1]). In particular, c2 (A) is monodromy invariant, under the diagonal action of the monodromy group of M on H 4 (M × M, Z), and hence remains of Hodge type under every smooth K¨ ahler deformation of M . Associated to a K¨ ahler class ω on M is a twistor deformation π : X → P1ω , where P1ω is the smooth conic defined in the complex projective plane P(H 2,0 (M ) ⊕ H 0,2 (M ) ⊕ Cω) via the BeauvilleBogomolov-Fujiki pairing. The ω-slope-stability of A and the invariance of c2 (A) imply that the sheaf A is ω-stable hyperholomorphic in the sense of Verbitsky [V1], which means that it deforms to a reflexive sheaf of Azumaya algebras A over the fiber square X ×P 1ω X of the twistor family. We shall be only interested in reflexive sheaves A which additionally satisfy a technical condition on their singularities spelled out in [MM2, Condition 1.6], but which we do not state in full here. Conjecture 1.6. [MM2, Conj. 1.12] Let M be an irreducible homolorphic symplectic manifold, ω a K¨ ahler class on M , and E a reflexive ω-slope-stable hyperholomorphic sheaf on M with an isolated singularity. Assume that H 1 (X, E) = 0. Denote by (Xt , Et ), t ∈ P1ω , the twistor deformation of (M, E). Then H 1 (Xt , Et ) = 0, for all t ∈ P1ω . The above conjecture is a theorem of Verbitsky when E is locally free [V3, Cor. 8.1]. Theorem 1.7. Assume that Conjecture 1.6 holds. fΛ of triples (M, η, A) as (1) [MM2, Theorem 1.8] There exists a coarse moduli space M above which is a non-Huasdorff complex manifold. The period map fΛ → {x ∈ P[Λ ⊗ C] : (x, x) = 0, (x, x) > 0} Pe : M

given by (X, η, A) 7→ η(H 2,0 (M )) is a surjective local analytic isomorphism. (2) [MM2, Theorem 1.9] The restriction of the period map to each connected component f0 of M fΛ is generically injective in the following sense. When the Picard group of M Λ M is trivial, or cyclic generated by a class of non-negative Beauville-Bogomolov-Fujiki

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f0 is the unique point of M f0 in the corresponding degree, then a point (M, η, A) of M Λ Λ fiber of Pe.

f0 of M fΛ containing a point of the form (X [n] , η0 , A0 ), where X0 is a Fix a component M 0 Λ K3 surface with a trivial Picard group and A0 is the modular Azumaya algebra of the Hilbert [n] scheme X0 . The modular Azumaya algebra A0 is ω-slope-stable, with respect to every K¨ ahler [n] [n] class ω on X0 × X0 , by [Ma7]. Denote by (1.11)

f0 Hilb ⊂ M Λ

the subset consisting of triples (X [n] , η, A), where X is a K3 surface with a trivial Picard group, X [n] is its Hilbert scheme, and A is the modular Azumaya algebra. By a Zariski open subset of an analytic space we mean the complement of a closed analytic subset. In view of Proposition 1.4, the following theorem is the deformation result mentioned above. Theorem 1.8. Assume that Conjecture 1.6 holds. There exists a Zariski dense open subset f0 , containing Hilb, a universal family π : M → U of irreducible holomorphic symplectic U ⊂M Λ manifolds, and a Brauer class Θ of order 2n − 2 over the fiber square M2 := M ×U M with the following properties. (1) The triple hF , ǫ, δi, given in Equation (1.10), deforms to a comonad object1 L := hF , ǫ, δi in D b (M2 , Θ). (2) Given an open subset V of U , denote by M2V the restriction of M2 to V and by ΘV the restriction of Θ to M2V . Define MV and F V similarly. When V is contractible and Stein, there exists a Brauer class θ over MV , such that ΘV = π1∗ θ −1 π2∗ θ, so that F V induces an endo-functor of D b (MV , θ). (3) The category of comodules D b (MV , θ)L carries a 2-triangulated structure2 such that the forgetful functor D b (MV , θ)L → D b (MV , θ) is 2-exact. The same holds for the category D b (Mu , θu )L u of comodules in D b (Mu , θu ) for the fiber Mu of π over each point u in U . (4) D b (Mu , θu )L u is a K3-category in the sense that the shift [2] is a Serre functor. (5) The open set U is large in the following sense. Let M be an irreducible holomorphic symplectic manifold of K3[n] -type, whose Picard group has rank ≤ 20. Then there exists an Azumaya algebra A over M × M , such that the triple (M, η, A) belongs to U .

Part 1 of the theorem is proven in section 6.1, part 2 in Remark 6.5, part 3 in section 6.2, part 4 in section 6.4, and part 5 in section 6.3. Let X be an algebraic K3-surface with Picard rank less than 20. In Section 7 we show that the deformations of X constructed in the above theorem via those of the Hilbert scheme X [n] may be interpreted infinitesimally as deformations of the category of coherent sheaves Coh(X) (see [To]). In fact, this family of deformations is the maximal family of generalized (non-commutative and gerby) deformations along which ideal sheaves of length n subschemes deform as objects of Coh(X). Similar statements are true of deformations coming from those of MH (v) provided the triple y = (MH (v), η, A), with A the modular Azumaya algebra of f0 . MH (v) (of Definition 1.5), belongs to M Λ 1 By a comonad object we mean that the convolutions F ◦ F ◦ · · · ◦ F are well defined and are all objects

of Db (M2 , Θ). Furthermore, the counit ǫ : F → O∆M and comultiplication δ : F → F ◦ F satisfy the axioms of a comonad. 2A 2-triangulated category is an additive category satisfying all the axioms of a Verdier triangulated category, except the octahedral axiom. See §6.2.

INTEGRAL TRANSFORMS AND DEFORMATIONS

9

There are natural homomorphisms HH 2 (X)

U◦

/ HomX×M (U , U [2]) o

◦U

HH 2 (M )

from the Hochschild cohomologies of X and M . The left homomorphism is an injection, while the right is an isomorphism. Inverting the right arrow and composing defines a homomorphism: φHH : HH 2 (X) → HH 2 (M ).

Set HT 2 (X) := H 0 (∧2 T X) ⊕ H 1 (T X) ⊕ H 2 (OX ) and similarly for M . Conjugating φHH with the Hochschild-Kostant-Rosenberg isomorphisms yields a map φT : HT 2 (X) → HT 2 (M ). For any class t ∈ HT 2 (X), let Coh(X, t) denote the first-order deformation of the category of coherent sheaves of X in the direction t (see the general construction by Toda [To]). Given a tangent vector ξ at a point u in the open subset U of Theorem 1.8, denote by Mξ the first order deformation of the fiber Mu over the length 2 subscheme of U corresponding to ξ. Let Lξ be the restriction of the comonad data L to Mξ . Recall that Mukai vectors of objects of D b (X) are naturally elements of HH0 (X) and that the Hochschild homology HH∗ (X) is an HH ∗ (X)-module [C2]. Let ann(v ∨ ) ⊂ HT 2 (X) be the image via the HKR-isomorphism of the subspace of HH 2 (X) annihilating the dual of the Mukai vector v. Theorem 1.9. Keep the notation of Theorem 1.1 and assume that the morphism α in Theorem 1.1 is an isomorphism. (1) The image φT (ann(v ∨ )) is the following subspace of HT 2 (M ): {(ξ, θ) : ξ ∈ H 1 (T M ), θ ∈ H 2 (OM ), and ξ · c1 (α) + (2 − 2n)θ = 0}. T

(2) Let φ : HT 2 (X) → H 1 (T M ) be the composition of φT with the projection HT 2 (M ) → T H 1 (T M ). Then φ restricts to ann(v ∨ ) as an isomorphism onto H 1 (T M ). Fix a class T t ∈ ann(v ∨ ) and set ξ := φ (t). The comonad category D b (Mξ )L ξ of Theorem 1.8 (3) is a triangulated category equivalent to the derived category D b (Coh(X, t)). 1.5. Notational conventions. We shall be working throughout over the complex numbers. The spaces that we deal with will be denoted by roman letters, while their deformations will be denoted by the corresponding calligraphic letters. For instance, if X is a K3 surface, then X will stand for a flat family with X as its central fiber. Azumaya algebras will be denoted by fraktur letters, such as A. Sheaves, and more generally, complexes of sheaves in the derived category will be denoted by script letters, while their deformations will be denoted by the same letter decorated with an over-line: for example, E and E . The same notational convention for deformations will be followed for Azumaya algebras and for morphisms between complexes. Given schemes or analytic spaces X and Y , and a morphism f : X → Y , we denote by f ∗ : D b (Y ) → D b (X) the left derived functor of the pullback functor from Coh(Y ) to Coh(X). When f is proper f∗ : D b (X) → D b (Y ) will denote the right derived functor of the direct image functor. Occasionally, we will use the notation Lf ∗ and Rf∗ for the same functors to emphasize their derived nature. Acknowledgments: We thank Ivan Mirkovic for pointing out to us the relevance of the categorical concept of monads. The work of Eyal Markman was partially supported by a grant from the Simons Foundation (#245840), and by NSA grant H98230-13-1-0239. Sukhendu Mehrotra was partially supported by a grant from the Infosys Foundation, and FONDECYT grant 1150404. He thanks Daniel Huybrechts for a useful conversation, to Emanuele Macr`ı for kindly inviting him to the Hausdorff Institute, and the institute for its hospitality. He is grateful to Paul Balmer and Matthias K¨ unzer, and especially to Moritz Groth for very useful email correspondence about N -triangulations.

10

E. MARKMAN AND S. MEHROTRA

2. A universal monad in D b (X × X) 2.1. The monad associated to a morphism of varieties. The following basic construction will be used repeatedly in the proof of the main result of this section. Construction 2.1. Let f : T → S be a morphism of smooth and projective varieties. We get the endofunctor f∗ f ∗ of D b (S), where the pullback and push forward are in the derived sense. Denote by u : 1S → f∗ f ∗ the unit for the adjunction, and let ǫ : f ∗ f∗ → 1T be the counit. Set µ := f∗ ǫf ∗ : f∗ f ∗ f∗ f ∗ → f∗ f ∗ and consider the monad Y := (f∗ f ∗ , u, µ) in D b (S). Any object, which is isomorphic to f∗ (G ) for some G ∈ D b (T ), admits an action (2.1)

f∗ ǫ

f∗ f ∗ f∗ G −→ f∗ G ,

so that the pair (f∗ G , f∗ ǫ) is an object of the category D b (S)Y of modules for the monad. We get the functor fe∗ : D b (T ) −→ D b (S)Y , sending an object G of D b (T ) to fe∗ (G ) := (f∗ G , f∗ ǫ). What is more, under the isomorphism f∗ f ∗ f∗ G → f∗ G ⊗ f∗ OT , this monadic action can be seen as an action of the algebra object f∗ OT . Indeed, recall that the product on f∗ OX is given by the composition µ

∼ OS = f∗ OT ⊗ f∗ OT −→ f∗ f ∗ f∗ f ∗ OS −→ f∗ f ∗ OS ∼ = f∗ OT .

The desired compatability between the the product on f∗ OX and its action on an object fe∗ (G ) := (f∗ G , f∗ ǫ) now follows from the axioms for a module for the monad Y.

2.2. A splitting of the monad. Keep the notation of Theorem 1.1. Set pt := Spec(C). Let c : M → pt be the constant map and set Y (OM ) := Rc∗ (OM ), as an object in D b (pt). Then Y (OM ) is naturally isomorphic to ⊕ni=0 Ext2i (OM , OM )[−2i], thought of as the Yoneda algebra of OM . ∗ ∗ ∗ → 1 Let u : π13∗ π13 X×X be the unit for the adjunction π13 ⊢ π13∗ , ǫ : π13 π13∗ → 1X×M ×X 2 ∗ ∗ the counit, and µ := π13∗ ǫ : (π13∗ π13 ) → π13∗ π13 the multiplication natural transformation. Denote by (2.2)

∗ Y := (π13∗ π13 , u, µ)

the monad in D b (X × X). We get the category D b (X × X)Y of modules for the monad Y and the functor b b Y πg 13∗ : D (X × M × X) → D (X × X) , as a special case of the construction in section 2.1. ∗ (U )∨ ⊗ π ∗ (U )[2] of D b (X × M × X) A is the push-forward of the object Af:= π1∗ ωX ⊗ π12 23 by π13 . We get a natural morphism (2.3)

m : A ⊗C Y (OM ) → A ,

so that the object (A , m) of D b (X × X)Y is the Y-module corresponding to the object Afvia the functor πg 13∗ . Note that the algebra structure on π13∗ OX×M ×X is now identified with cup product on H ∗ (OM ), or the composition product on Y (OM ) = ⊕ni=0 Ext2i (OM , OM )[−2i]. Let η : ∆∗ OX → A be the morphism corresponding to the unit of the adjunction ΦU ⊣ ΨU . We get the composite morphism q

(2.4)

∆∗ OX ⊗C Y (OM )

η⊗id

/ A ⊗C Y (OM )

m

' /A.

INTEGRAL TRANSFORMS AND DEFORMATIONS

11

n−1 Let λn be the object ⊕i=0 Ext2i (OM , OM )[−2i] in D b (pt). We have a natural morphism ι : λn → Y (OM ). The object Y (OM ) is naturally the direct sum of λn and Ext2n (OM , OM )[−2n]. Set α := q ◦ ι : ∆∗ OX ⊗C λn → A . So α is the composition

(2.5)

ι

η⊗id

m

∆∗ OX ⊗C λn −→ ∆∗ OX ⊗C Y (OM ) −→ A ⊗C Y (OM ) −→ A .

The main result of this section is Theorem 2.2. Let v ∈ K(X) be a primitive class with (v, v) = 2n − 2, n ≥ 2, and H a v-generic polarization. (1) When v = (1, 0, 1 − n), that is when M := MH (v) is the Hilbert scheme of n points on X, then the morphism α, displayed in equation (2.5), is an isomorphism. In particular, a choice of a non-zero element tM of Ext2 (OM , OM ) determines an isomorphism n−1 A ∼ ∆∗ OX [−2i]. = ⊕i=0

(2) In general, for v arbitrary, the structure sheaf of the diagonal ∆∗ OX is a direct summand of A in D b (X×X). In particular, the integral transform ΦU : D b (X) → D b (M, θ) is faithful. Part (1) of the theorem is proven in Section 2.3 and part (2) in Section 2.5. The splitting of the monad object A in Theorem 2.2 (1) extends over a Zariski open subset of the base of a family in the following sense. Let π : X → B be a smooth and proper family of K3 surfaces over an analytic space B and v a continuous primitive section of the local system Rπ∗ Z of Mukai lattices. Let p : M → B be a smooth and proper family of irreducible holomorphic symplectic manifolds, such that each fiber Mb of p is isomorphic to the moduli space MHb (vb ) of Hb -stable sheaves with Mukai vector vb over the fiber Xb of π for some polarization Hb over Xb . We do not assume that Hb varies continuously (see for example [Y1, Prop. 5.1]). There exists a twisted sheaf U over X ×B M , flat over B, such that its restriction Ub to Xb × Mb is a universal sheaf for the coarse moduli space MHb (vb ), by the appendix in [Mu5]. Denote by Ab the monad object over Xb × Xb associated to Ub . The construction of the morphism α, given in Equation (2.5) above, goes through in this relative setting to yield a family of morphisms αb : ∆∗ OXb ⊗C λn → Ab , b ∈ B, corresponding to a global morphism  n−1 2i α : ∆∗ OX ⊗OB ⊕i=0 R p∗ OM [−2i] → A of monad objects over X ×B X .

Lemma 2.3. The locus B0 in B, where αb is an isomorphism, is a Zariski open subset of B in the analytic topology. The open subset B0 is non-empty, whenever there exists a point b0 ∈ B, a K3 surface X, and an equivalence of derived categories D b (Xb ) → D b (X), which maps the Mukai vector vb to that of the ideal sheaf of a length n subscheme of X, and which maps Hb -stable sheaves on Xb to ideal sheaves on X. Proof. B0 is the complement of the image in B of the support of the object Cα in D b (X ×B X ) representing a cone of the morphism α. The support of Cα is the union of the support of the sheaf cohomologies of Cα . The support of Cα intersects the fiber Xb × Xb if and only if αb is not an isomorphism, since a point x ∈ Xb × Xb belongs to the support of Cα if and only if one of the the cohomologies of Cα ⊗ Ox is non-zero, by [BM, Lemma 5.3]. The non-emptiness statement follows from Theorem 2.2 (1).  A more explicit extension of the splitting Lemma 2.3 is given in Lemmas 6.9 and 6.16.

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E. MARKMAN AND S. MEHROTRA

2.3. Splitting of the monad in the Hilbert scheme case. Write X [n] for the Hilbert scheme of n points on X and S n X for the n-th symmetric product of X. Denote by µ the Hilbert-Chow morphism from X [n] to S n X, and let π : X n → S n X be the quotient by Sn . The following result is the main point of the proof of Theorem 2.2 for M = X [n] ; it allows us to transport calculations from the derived category of X [n] to the more combinatorial b (X n ) below. Sn -equivariant derived category of X n . The latter is denoted by DS n Theorem 2.4. ([Ha2], Corollary 5.1) Let X be a smooth quasi-projective surface, and denote by Bn the reduced fiber-product of X [n] and X n over S n X: ①① ①① ① ① {① ① q

X [n]●

●● ●● ● µ ●●● #

Bn ❊

❊❊ p ❊❊ ❊❊ ❊"

Xn

① ①① ①① ① ① |① π ①

SnX

∼ =

b (X n ) is an equivalence of derived Then, the map q is flat, and Rp∗ q ∗ : D b (X [n] ) −→ DS n categories.

Let us give a quick word of explanation here. Denote by HilbSn (X n ) the Sn -Hilbert scheme of X n , the fine moduli space whose closed points parametrize the Sn orbits in X n with structure sheaves isomorphic to the regular representation of Sn . The flatness of the map q above says that Bn ⊂ X n ×X [n] is a family of such Sn orbits parametrized by X [n] . This yields a morphism from X [n] to HilbSn (X n ), which is in fact seen to be an isomorphism, identifying Bn → X [n] with the universal family of HilbSn (X n ). On the other hand, given a finite group G acting nicely3 on a smooth projective variety M , the derived McKay correspondence of [BKR] relates the derived category of the G-Hilbert b (M ): whenever the map scheme D b (HilbG (M )) and the G-equivariant derived category DG G Hilb (M ) → M/G is semismall, it says that the structure sheaf of the universal family gives a Fourier-Mukai type equivalence between these two categories. The map µ : X [n] → S n X satisfies the semismallness hypothesis. This establishes the second statement of Theorem 2.4. The following is a special case of a vanishing theorem proved in [Ha2]. Let Zn ⊂ X × Bn be the pullback of the universal family Un ⊂ X × X [n] of n points on X, and Dn ⊂ X × X n the union of graphs of the n projections to the ith factor, πi : X n → X. Consider the following diagram t / Bn Zn ❑ ❍❍ s ❑ s ❑ ❍❍ s ❑ ❑❑ ❍❍ ss ∩ s ❑ ❍❍ s ❑ ❑❑ ss ❍❍  s ❑ s ❍❍p ❑ s ❑ s ❍❍ ❑ X × B ❑❑ n ss ❍❍ ▲ s ❑ q ▲ s q ❑ ▲ ❍❍ s q ▲▲id×p=ˆp ❑❑ id×q=ˆ qqq ❍❍ ss ▲ ❑ q s ▲ ❑ q ❍❍ s ▲ q ❑ s ▲ q ❑ s ❍$ q x ▲ ❑% % yss ⊂ ⊃ s [n] n o / / X ×X X ×X Dn Xn Un

Theorem 2.5 ([Ha2], Proposition 5.1). Rˆ p∗ OZn ∼ = ODn . 3Nicely here means that the canonical bundle of M is locally trivial as a G-sheaf.

INTEGRAL TRANSFORMS AND DEFORMATIONS

13

Proposition 2.6. Rˆ p∗ qˆ∗ (IUn ) ∼ = IDn , where IUn is the ideal sheaf of Un ⊂ X × X [n] ,and n IDn that of Dn ⊂ X × X .

Proof. Applying Rˆ p∗ qˆ∗ to the sequence

0 → IUn → OX×X [n] → OUn → 0,

and using the previous result, we obtain the exact triangle

Rˆ p∗ IZn → Rˆ p∗ OX×Bn → ODn . Thus, it suffices to show that Rˆ p∗ OX×Bn ∼ = OX×X n , or even that Rp∗ OBn ∼ = OX n . Now, t [n] Zn → Bn is flat and finite, being the pullback of Un → X . Therefore, t∗ OZn contains OBn s as a direct factor. Further, Dn → X n is finite, and s ◦ pˆ = p ◦ t, so that s∗ Rˆ pOZn ∼ = s∗ ODn ∼ = Rp∗ t∗ OZn is concentrated in degree 0. Consequently Rp∗ OBn is concentrated in degree 0 also, and because p has connected fibers, we are done. (See also Prop. 1.3.3, [Sc].)  Lemma 2.7.

(i) Suppose the schemes and morphisms in the commutative diagram ⑥⑥ ⑥⑥ ⑥ ⑥ ~⑥ ⑥ Y ❆ ❆❆ ❆❆ ❆ µ ❆❆ q

X❆

W

❆❆ ❆❆p ❆❆ ❆

Z ⑥⑥ ⑥⑥ ⑥ ⑥ ~⑥ π ⑥

are such that Lq ∗ and p! are defined between bounded derived categories (for example, if Y, Z are smooth and projective, and X is a closed subscheme of Y × Z). If Φ := ∼ = Rp∗ Lq ∗ : D b (Y ) −→ D b (Z) is an equivalence, given M , N ∈ D b (Y ), we have a bifunctorial isomorphism µ∗ RH om b (M , N ) ∼ = π∗ RH om b (Φ(M ), Φ(N )). D (Y )

D (Z)

(ii) Suppose G is a finite group, and the morphism p : X → Z in part (i) is G-equivariant. Further, let Y = X/G, W = Z/G, q and π the quotient morphisms, and µ : X/G → b (Z) the derived category of equivariant Z/G the morphism induced by p. Denote by DG ∼ =

b (Z) is an equivalence, there coherent sheaves. Then, if Φ := Rp∗ Lq ∗ : D b (Y ) −→ DG is a bifunctorial isomorphism µ∗ RH omDb (Y ) (M , N ) ∼ = [π∗ RH omDb (Z) (Φ(M ), Φ(N ))]G . G

Proof. Both parts follow from essentially the same formal calculation using Grothendieck Duality. We provide a proof of part (ii). Denote by π∗G the composition [−]G ◦ π∗ ; the symbols q∗G , µG ∗ are defined similarly. Then, (q ∗ M , p! p∗ q ∗ N ) π G RH om b (p∗ q ∗ M , p∗ q ∗ N ) ∼ = π G p∗ RH om b ∗

DG (Z)



DG (X)

∼ = µ∗ q∗G RH omDGb (X) (q ∗ M , p! p∗ q ∗ N ) ∼ = µ∗ [RH omDb (Y ) (M , q∗ p! p∗ q ∗ N )]G ∼ = µ∗ RH om b (M , (q G p! )(p∗ q ∗ )N ) D (Y )



∼ = µ∗ RH omDb (Y ) (M , N ).

The first isomorphism is Grothendieck Duality, the second follows from the commutativity of the diagram above and the G-invariance of µ, the third is adjunction, and the fourth follows

14

E. MARKMAN AND S. MEHROTRA

from the G-invariance of M (see [Sc], Prop. 1.3.2). The last isomorphism follows from the  fact that q∗G p! is the right adjoint of Φ = p∗ q ∗ , and so also its quasi-inverse. X × Bn ×❘X

❧❧❧ ❧❧❧ ❧ ❧ ❧ v ❧❧ ❧ q

❘❘❘ ❘❘❘p ❘❘❘ ❘❘❘ (

X × X [n]❊×❘ X

X × Xn × X

❧ ❊❊❘❘❘❘ ❧❧❧ ②② ❊❊ ❘❘❘ ❧❧❧ ②②② ❧ ❊❊ µ ❘❘❘❘ ❧ ❧❧ π ②②② ❊❊ ❘❘) v❧❧❧ ❊❊ ② n ❊ ②② π13 ❊❊❊X × S X × X ②②② p13 ❊❊ ②② ❊❊ f ②② ❊❊ ② "  |②②

X ×X

We shall now apply this lemma to the diagram above: Let pij stand for the projection from the product X × X n × X to the (i, j)-th factor. Consider the object A , given in equation (1.2), in the case M = X [n] and U = IUn . In view of Proposition 2.6, we immediately have the isomorphism A ∼ = {Rp13∗ (p∗ I ∨ ⊗ p∗ ID )[2]}Sn . 12

Dn

23

n

Furthermore, the above is an isomorphism of Y (OX [n] )-modules, under the natural identification of Y (OX [n] ) with Y (OX n )Sn . Denote by ∆i ⊂ X × X n , the graph of the i-th projection πi : X n → X, and by ∆I , ˇ for I ⊂ 1, ..., n, the partial diagonal ∩i∈I ∆i . By Corollary A.4 of [Sc], there is a Cech-type Sn -equivariant resolution of IDn as follows: 0 → IDn → OX×X n → ⊕ni=1 O∆i → · · · → ⊕|I|=k O∆I → · · · → O∆{1,...,n} → 0. As the diagonals ∆i intersect transversally, it is easy to see that, in fact, this resolution is the tensor product of complexes ⊗ni=1 {OX×X n → O∆i }, or alternatively, I Dn ∼ = I ∆1 ⊗ I ∆2 ⊗ · · · ⊗ I ∆n

b (X × X n ), where the I in DS ∆i are the ideal sheaves of the indicated diagonals. n

Remark 2.8. The Sn -linearization of the component ⊕|I|=k O∆I in the complex above consists simply of permuting the factors O∆I according to the action of Sn on the indexing sets I. Tracing through the above calculation, it is easily seen that the Sn -linearization of (I∆1 ⊗ I∆2 ⊗ · · · ⊗ I∆n ) is also the expected one, namely, permutation of factors. The following calculation is due to Mukai. We present the details for the convenience of the reader Lemma 2.9. ([Mu5], Prop. 4.10). Let pij be the (i, j)-th projection from X × X × X. Then, there is a natural isomorphism B := p13,∗ (p∗12 I∆∨ ⊗ p∗23 I∆ ) ∼ = H 2 (X, OX )⊗C O∆ [−2], where ∆ is the diagonal in X × X. Equivalently, the relative extension sheaves E xtjp13 (p∗12 I∆ , p∗23 I∆ ) vanish, for j 6= 2, and E xt2p13 (p∗12 I∆ , p∗23 I∆ ) ∼ = H 2 (X, OX ) ⊗C O∆ .

INTEGRAL TRANSFORMS AND DEFORMATIONS

15

Proof. The convolution of the exact triangle ∨ O∆ → OX×X → I∆∨

with I∆ yields the following exact triangle on X × X: p13∗ RH om(p∗12 O∆ , p∗23 I∆ ) → p13,∗ p∗23 I∆ → B. By the use of flat base-change for the Cartesian diagram X × X ×❖X

♣ ♣ ♣♣ w ♣♣ ♣ X × X❖ ❖❖❖ ❖❖❖ p1 ❖❖❖❖ ❖'

❖❖❖ ❖❖p❖13 ❖❖❖ ❖'

p12 ♣♣♣♣

X ×X

X

♦♦ ♦♦♦ ♦ ♦ ♦♦ p1 w ♦♦ ♦

the second term in the triangle is isomorphic to p∗1 p1∗ I∆ . It is then simple to conclude from the short exact sequence 0 → I∆ → OX×X → O∆ → 0 ∼ p∗ p1∗ I∆ ∼ that p13,∗ p∗23 I∆ = = H 2 (X, OX ) ⊗C OX×X [−2]. The first term in the triangle is 1 computed as follows: p13∗ RH om(p∗12 O∆ , p∗23 I∆ ) ∼ = p13∗ RH om(∆12∗ OX×X , p∗23 I∆ ) ∼ = p13∗ ∆12∗ RH om(OX×X , ∆! p∗ I∆ ) 12

23

−1 ∼ ⊗ ∆∗12 p∗23 I∆ [−2]) = p13∗ ∆12∗ RH om(OX×X , p∗3 ωX ∼ = p∗ ω −1 ⊗ I∆ [−2]. 2 X

The first isomorphism is flat base-change, while the second is Grothendieck duality. As Hom(I∆ , OX×X ) ∼ = C and B is supported on the diagonal, it follows that B is isomorphic 2 to H (X, OX ) ⊗C O∆ [−2].  Let us now calculate Rp13∗ (p∗12 ID∨n ⊗ p∗23 IDn ) when n = 2; the answer for general n will be apparent from this. Set X1 = · · · X4 = X, and for I ⊂ {1, ..., 4}, let XI := ×i∈I Xi . Denote by uI : X{1,..,4} → XI the projection to the I-th factor; similarly, for I ⊂ J ⊂ {1, .., 4}, |I| = 2, |J| = 3, let vIJ : XJ → XI be the obvious projection. X1 × X2 ❥❥ u124❥❥❥❥❥

❥❥❥❥ u ❥❥❥ ❥

X1 × X2 × X ❚4

❚❚❚❚ ❚❚❚❚ ❚❚❚❚ 124 ❚❚) v14

× X3 ❚× X4

❚❚❚❚ ❚❚u❚134 ❚❚❚❚ ❚❚❚)

X × X3 × X4

u14



X1 × X4

1 ❥❥❥❥ ❥ ❥ ❥ ❥❥❥❥134 u ❥❥❥ v14 ❥

16

E. MARKMAN AND S. MEHROTRA

We then have Rp13∗ (p∗12 ID∨2 ⊗ p∗23 ID2 ) ∼ = u14∗ {u∗12 I∆∨ ⊗ u∗13 I∆∨ ⊗ u∗24 I∆ ⊗ u∗34 I∆ } ∼ v 134 u134∗ {(u∗ v 134∗ I ∨ ⊗ u∗ v 134∗ I∆ ) ⊗ (u∗ I ∨ ⊗ u∗ I∆ )} = 14∗ 134 13 ∆ 134 34 12 ∆ 24 134 134∗ ∨ 134∗ ∗ ∨ ∗ ∼ I ⊗v I∆ ) ⊗ u134∗ (u I ⊗ u I∆ )} = v {(v 14∗

13



34

12



24

134 134∗ ∨ 134∗ 124∗ ∨ 124∗ ∼ {(v13 I∆ ⊗ v34 I∆ ) ⊗ u134∗ u∗124 (v12 I∆ ⊗ v24 I∆ )} = v14∗ ∼ = v 134 {(v 134∗ I ∨ ⊗ v 134∗ I∆ ) ⊗ v 134∗ v 124 (v 124∗ I ∨ ⊗ v 124∗ I∆ )} 14∗

13



34

14

14∗

12



24

134 134∗ ∨ 134∗ 124 124∗ ∨ 124∗ ∼ (v13 I∆ ⊗ v34 I∆ ) ⊗ v14∗ (v12 I∆ ⊗ v24 I∆ ) = v14∗   2 2 ∼ = H (X, OX ) ⊗C O∆ [−2] ⊗ H (X, OX ) ⊗C O∆ [−2] ,

where the third isomorphism is the projection formula, the fifth is flat base-change and the last one is the isomorphism of the previous lemma. Clearly, the same working proves the n⊗ as objects in the Proposition 2.10. Rp13∗ (p∗12 ID∨n ⊗ p∗23 IDn ) ∼ = H 2 (X, OX ) ⊗C O∆ [−2] b (X × X) (where S -acts trivially on X × X). Moreover, the S -linearization of category DS n n n the tensor product on the right hand side is simply permutation of factors. Proof. The linearization is clear from the calculation above and Remark 2.8.



Lemma 2.11. Let S be a scheme and F an object in D b (S). Assume that the cohomology sheaves Hj (F ) satisfy the condition: Extk+1 (Hj (F ), Hj−k (F )) = 0, ∀j and ∀k > 0. Then F is isomorphic to ⊕j Hj (F )[−j]. Proof. Set JF := {j : Hj (F ) 6= 0}. The proof is by induction on the cardinality ♯(JF ) of JF . Set b := max(JF ) and a := min(JF ). Then F is represented by a complex of coherent OS -modules Fa → Fa+1 → · · · → Fb ,

with Fi in degree i. If ♯(JF ) = 1, then a = b and the statement holds trivially. Assume that the statement holds for every object F ′ satisfying ♯(JF ′ ) < n, where n := ♯(JF ). Set A := Ha (F ). Let F ′ be the cone of the natural morphism Ha (F )[−a] → F . Then ♯(JF ′ ) = n−1, Ha (F ′ ) = 0, and Hi (F ′ ) = Hi (F ), for i 6= a. The equivalence F ′ = ⊕j Hj (F ′ )[−j] follows, by the induction hypothesis. We get the exact triangle δ

A[−a] → F → F ′ → A[1 − a]. The morphism δ decomposes as the sum of the morphisms δj ∈ Hom(Hj (F ′ )[−j], A[1 − a]) = Ext1+j−a (Hj (F ), Ha (F )), for j > a. These groups vanish, by assumption. Hence, δ = 0, and F is isomorphic to the direct sum of A[−a] and F ′ .  Proof of Theorem 2.2 part 1. Step 1: To compute the Sn -invariants in Rp13∗ (p∗12 ID∨n ⊗ p∗23 IDn ), we first calculate the Sn -invariant parts of the the n-fold multi-tors T orqn (O∆ ) := T orq (O∆ , O∆ , · · · , O∆ ). This is precisely the sort of calculation that is carried out in Corollary B.7 in [Sc]. In our particular situation this says that T orqn (O∆ ) has nonzero Sn -invariants if and only if q = 2h, 0 ≤ h ≤ n − 1, in which case 2

n ∗ T or2h (O∆ )Sn ≃ (∧ N∆/X×X )⊗h ≃ H 0 (X, ωX )⊗h ⊗C O∆ .

INTEGRAL TRANSFORMS AND DEFORMATIONS

17

∗ I , π ∗ I ) are isomorphic to ∆ O , for Equivalently, the relative extension sheaves E xtj (π12 ∗ X U 23 U j even in the range 2 ≤ j ≤ 2n, and vanish for all other values of j. ∗ I ∨ ⊗ π ∗ I ), are satisfied, The hypotheses of Lemma 2.11, applied to the object Rπ13∗ (π12 23 U U since the odd cohomologies of this object vanish, and ExtiX×X (O∆ , O∆ ) = 0 for odd i. Lemma 2.11 implies the existence of an isomorphism (2.6) Rπ13∗ (π ∗ I ∨ ⊗ π ∗ IU ) ∼ = ⊕n H 2 (X, OX )⊗i ⊗C ∆∗ OX [−2i]. 12

U

23

i=1

Note that the i-th summand on the right hand side is naturally isomorphic to the sheaf coho∗ ∗ mology of degree 2i of the left hand side, namely the relative extension sheaf E xt2i π13 (π12 IU , π23 IU ). n−1 2 The isomorphism A ∼ H (X, OX )⊗i ⊗C ∆∗ OX [−2i] follows immediately from Equation = ⊕i=0 (2.6). In particular, the functor ΦU is faithful. Step 2: We prove in this step the following claim. Claim 2.12. For i even, we have the canonical short exact sequence  fi  ei i+2 0 → H i (A ) → ∆X∗ ωX ⊗ E xti+2 (A ) ⊗ π1∗ ωX → 0. πX (IU , IU ) → H For i odd, we have the following natural isomorphism: h i fi (I , I ) −→ H i+1 (A ) ⊗ π1∗ (T ∗ X). ∆X∗ ωX ⊗ E xti+2 U U π X∗

Proof. Let ∆13 : X × M → X × M × X be the diagonal embedding. Consider the short exact sequence: ∗ 0 → π13 I∆X → OX×M ×X → ∆13∗ OX×M → 0. ∗ f Tensoring with A := π ωX [2] ⊗ π ∗ I ∨ ⊗ π ∗ IU we get the exact triangle: 1

(2.7)

12

U

23

 f˜ e˜ ∗ ∗ I∆X [1] Af⊗ π13 I∆X → Af→ π1∗ ωX [2] ⊗ ∆13∗ IU∨ ⊗ IU → Af⊗ π13 g˜

Applying the derived push-forward π13∗ to the exact triangle (2.7) we get the following exact triangle in D b (X × X).   f g e (2.8) A ⊗ I∆X → A → ∆X∗ ωX [2] ⊗ πX∗ IU∨ ⊗ IU → A ⊗ I∆X [1].

Let ι : I∆X → OX×X be the natural inclusion. The object A has been shown to be a direct sum of sheaves supported on the diagonal, and the morphism g is the composition 1 ⊗ι A ⊗ I ∆X A → A ⊗ OX×X ∼ = A . Hence, the morphism g vanishes and we get a short exact sequence, for every integer i.  f i  ei 0 → H i (A ) → ∆X∗ H i ωX [2] ⊗ πX∗ IU∨ ⊗ IU → H i+1 (A ⊗ I∆X ) → 0. It remains to calculate the sheaf cohomologies H i+1 (A ⊗ I∆X [1]). Note first the isomorphisms Tor0X×X (O∆X , I∆X ) ∼ = ∆X∗ T ∗ X, ∼ ∆X ω X , TorX×X (O∆ , I∆ ) =

∗ X X −1 X×X and Tori (O∆X , I∆X ) = 0, if i is not equal to 0 or −1. We have already shown that n−1 isomorphic to ⊕i=0 H 2i (A ). For even i we get the isomorphism  ∼ ∼ TorX×X H i+2 (A ), I∆ H i+1 (A ⊗ I∆X ) = = H i+2 (A ) ⊗ π1∗ ωX . X −1

For odd i we get:

H i+1 (A ⊗ I∆X ) ∼ = Tor0X×X H i+1 (A ), I∆X



∼ = H

i+1

(A ) ⊗ π1∗ (T ∗ X).

A is



18

E. MARKMAN AND S. MEHROTRA

∗ ∗ Step 3: We have the isomorphisms H 2i (A ) ∼ = π1∗ ωX ⊗E xt2i+2 π13 (π12 IU , π23 IU ), by definition of A . The homomorphisms  ∗ ∗ ej−2 : E xtjπ13 (π12 IU , π23 IU ) → ∆∗ E xtjπX (IU , IU )

are injective, for all j, and e2n−2 is an isomorphism, by Claim 2.12. Let tM be a non-zero ele∗ I ∨ ⊗ π∗ I ) ment of H 2 (M, OM ), where M = X [n] . Then tM induces a morphism from π13∗ (π12 23 U U ∗ ∨ ∗ to π13∗ (π12 IU ⊗ π23 IU ) [2]. We get an induces homomorphism of relative extension sheaves ∗ ∗ ∗ ∗ tM : E xtjπ13 (π12 IU , π23 IU ) → E xtj+2 π13 (π12 IU , π23 IU ) ,

via the morphism (2.3). We prove in this step the following statement. Claim 2.13. The sheaf homomorphism (2.9)

∗ ∗ tiM : E xt2π13∗ (π12 IU , π23 IU )

∗ ∗ E xt2i+2 π13∗ (π12 IU , π23 IU )



is an isomorphism, for 0 ≤ i ≤ n − 1. Proof. We have the homomorphism µIU (tM ) : E xtjπX∗ (IU , IU ) → E xtj+2 πX∗ (IU , IU ) . µI

tr

U The composition OX×M −→ IU∨ ⊗ IU −→ OX×M is the identity, since rank(IU ) = 1. Thus, the composition

µI

U Rπ0 X∗ OX×M −→ E xt0πX (IU , IU )

µIU (tM )n

−→

tr

2n E xt2n πX (IU , IU ) −→ RπX∗ OX×M

is multiplication by tnM . Hence, the following is an isomorphism µIU (tM )n : E xt0πX∗ (IU , IU )

−→

E xt2n πX∗ (IU , IU ) .

The exact triangle (2.8) is the image of one in D b (X × M × X) and lifts to an exact triangle in the category D b (X × X)Y for the monad Y given in (2.2). Taking cohomology we get the following commutative diagram: ∗ ∗ E xt2i π13∗ (π12 IU , π23 IU )

tM

∗ ∗ E xt2i+2 π13∗ (π12 IU , π23 IU )

/

e2i−2

e2i



  ∆X∗ E xt2i πX (IU , IU )

∆∗ (µIU (tM ))



π1∗ 1ωX ⊗tM



  ∆X∗ E xt2i+2 πX (IU , IU ) /

f 2i

f 2i−2

∗ ∗ π1∗ ωX ⊗ E xt2i+2 π13∗ (π12 IU , π23 IU )

/



∗ ∗ π1∗ ωX ⊗ E xt2i+4 π13∗ (π12 IU , π23 IU ) .

We are ready to prove that the homomorphism (2.9) is an isomorphism. The proof is by contradiction. Assume otherwise and let i be the minimal integer in the range 1 ≤ i ≤ n − 1, such that (2.10)

∗ ∗ tiM : E xt2π13∗ (π12 IU , π23 IU )



∗ ∗ E xt2i+2 π13∗ (π12 IU , π23 IU )

is not an isomorphism. Then the above homomorphism must vanish. Hence, ∗ ∗ IU , π23 IU )⊗ π1∗ ωX (2.11) tiM ⊗ π1∗ 1ωX : E xt2π13∗ (π12



∗ ∗ ∗ E xt2i+2 π13∗ (π12 IU , π23 IU ) ⊗ π1 ωX

INTEGRAL TRANSFORMS AND DEFORMATIONS

19

vanishes as well. Consider the following (abreviated) commutative diagram with short exact columns. / E xt2 π13

0 e−2

tM

e0



∆X∗ E xt0πX

/ ∆X E xt2 ∗ πX

f −2 ∼ =

/ ···



/ ∆X E xt2i ∗ πX

f0



E xt2π13 ⊗ π1∗ ωX

tM ⊗π1∗ 1ωX

/ E xt2i+2 π13

µIU (tM )



e2i

e2i−2



µIU (tM )

tM

/ E xt2i π13

/ ···

/ ∆X E xt2i+2 ∗ πX

f 2i−2



/ E xt4 ⊗ π ∗ ωX π13 1

/ ···



/ E xt2i+2 ⊗ π ∗ ωX π13 1

The vanishing of (2.11), the fact that f −2 is an isomorphism, and the exactness of the columns in the above diagram, combine to imply that the image of ∆X∗ (µIU (tM ))i : ∆X∗ E xt0πX (IU , IU ) → ∆X∗ E xt2i πX (IU , IU )

is contained in the image of the injective homomorphism

∗ ∗ 2i e2i−2 : E xt2i π13 (π12 IU , π23 IU ) → ∆X∗ E xtπX (IU , IU ) .

Hence, the injective homomorphism

∆X∗ (µIU (tM ))i+1 : ∆X∗ E xt0πX (IU , IU ) → ∆X∗ E xt2i+2 πX (IU , IU )

factors through

∗ 2i+2 ∗ ∗ ∗ tM : E xt2i π13 (π12 IU , π23 IU ) → E xtπ13 (π12 IU , π23 IU ) .

In particular, the above homomorphism does not vanish. Hence, the above homomorphism is an isomorphism. The minimality of i implies that ∗ ∗ ∗ ∗ i−1 : E xt2π13 (π12 IU , π23 IU ) → E xt2i tM π13 (π12 IU , π23 IU )

is an isomorphism. It follows that the homomorphism (2.10) is an isomorphism. A contradiction. This complete the proof of Claim 2.13.  ∗ I , π ∗ I ) → Rπ (π ∗ I ∨ ⊗ π ∗ I ) be the natural morphism Step 4: Let ι2 : E xt2π13 (π12 13 U 23 U 12 U 23 U from the first non-vanishing sheaf cohomology of the object to the object itself. Let   ∗ ∗ ∗ ∗ IU∨ ⊗ π23 IU IU∨ ⊗ π23 IU ⊗C Y (OM ) → Rπ13 π12 m : Rπ13 π12

be the natural morphism. The morphism of Y (OM )-modules

∗ ∗ ∗ ∗ m ◦ (ι2 ⊗ id) : E xt2π13 (π12 IU , π23 IU ) ⊗C Y (OM ) → Rπ13 π12 IU∨ ⊗ π23 IU



induces an isomorphism of the sheaf cohomologies for degrees between 2 and 2n, by Claim 2.13. Thus, the composite morphism α in equation (2.5) induces an isomorphism of all sheaf cohomologies, i.e., for degrees in the range from 0 to 2n − 2.  2.4. Hochschild (co)homology. We present a brief review here of some concepts and definitions on this topic which will be used in the proof of part 2 of Theorem 2.2, in the proof of Proposition 3.16, and also later in Section 7. We follow the presentation in [C2, CW]. Those familiar with the Hochschild (co)homology of varieties should skip to Section 2.5. Let T be a smooth, projective variety over C. Write ∆T : T → T × T for the diagonal, and πi : T × T → T, i = 1, 2, for the projections; denote by S∆T the kernel of the Serre functor −1 the object ∆T,∗ ωT−1 [− dim T ]. Let ∆T,! : D b (T ) → D b (T × T ) be ∆T,∗ ωT [dim T ], and by S∆ T the left adjoint of ∆∗T .

20

E. MARKMAN AND S. MEHROTRA

Definition 2.14. (i) The Hochschild structure of T consists of a graded ring HH ∗ (T ), HH i (T ) := HomT ×T (O∆T , O∆T [i]), and a graded left module HH∗ (T ) over HH ∗ (T ), −1 [i], O∆T ). HHi (T ) := HomT ×T (∆T,! OT [i], O∆T ) = HomT ×T (S∆ T

Both the ring and module structures are defined by Yoneda composition in D b (T × T ). (ii) The harmonic structure of T consists of a graded ring HT ∗ (T ), M HT i (T ) := H p (T, ∧q TT ), p+q=i

and a graded module HΩ∗ (T ) over HT ∗ (T ), HΩi (T ) :=

M

H p (T, ΩqT ).

q−p=i

Exterior product and contraction define the ring and module structures, respectively. These two structures are related by the Hochschild-Kostant-Rosenberg isomorphism M ∼ (2.12) IT : ∆∗T O∆T −→ ΩiT [i], i

which yields isomorphisms of graded vector spaces: ∼

I ∗ : HT ∗ (T ) −→ HH ∗ (T ), ∼

I∗ : HH∗ (T ) −→ HΩ∗ (T ). We shall also have occasion to work with the following modified isomorphisms p ∗ 1 ∗ T = ( td ∧ ( )) ◦ I . y( )), and If (2.13) If T T = I ◦ (√ ∗ ∗ tdT Hochschild homology behaves functorially under integral transforms: associated to an integral functor Φ : D b (T ) → D b (Y ), there is a natural map of graded vector spaces (2.14)

Φ∗ : HH∗ (T ) → HH∗ (Y ),

which is constructed as follows. Let F be the kernel of Φ, R the kernel of its right adjoint, and L the kernel of its left adjoint. Denote by pij the projection from T × Y × T to its ij-th factor. Any object Q ∈ D b (Y × T ) gives rise to a functor D b (T × T ) Q ◦ : D b (T × Y ) → T 7→ p13,∗ (p∗12 (T ) ⊗ p∗23 (Q))

(2.15)

via convolution; define ◦ Q : D b (Y × T ) → D b (Y × Y ) similarly. We first define a natural map Φ† : HomY ×Y (O∆Y , S∆Y [i]) → HomT ×T (O∆T , S∆T [i]). Given a morphism ν : O∆Y → S∆Y [i], let Φ† ν be the composite morphism (2.16)

η

ν

ǫ

O∆T → R ◦ F = R ◦ O∆Y ◦ F → R ◦ S∆Y [i] ◦ F = S∆T [i] ◦ L ◦ F → S∆T [i],

where η and ǫ are the natural unit and the counit, respectively. Notice that we have the isomorphism (2.17) HomT ×T (O∆ , S∆ [i]) ∼ = HHi (T )∨ T

T

INTEGRAL TRANSFORMS AND DEFORMATIONS

21

by Serre duality on T × T , and that HomY ×Y (O∆Y , S∆Y [i]) ∼ = HHi (Y )∨ in the same way. The above construction defines a homomorphism Φ† : HHi (Y )∨ → HHi (T )∨ , ν 7→ Φ† ν.

Then the desired map Φ∗ is the transpose of Φ† under these identifications. Recall that any integral transform Φ induces a map ϕ : H ∗ (T, C) → H ∗ (Y, C) on singular cohomology. We have the following result stating the compatibility of this map with Φ∗ under the HKR isomorphism: Theorem 2.15 ([MSM]). Let Φ : D b (T ) → D b (Y ) be an integral transform. Then, the following diagram commutes. HH∗ (T ) Ie∗T

Φ∗





HΩ∗ (T )

/ HH∗ (Y )

ϕ

Ie∗Y

/ HΩ∗ (Y )

2.5. Splitting of the monad for a general moduli space. Let v ∈ H ∗ (X) be a primitive and effective class with hv, vi = 2n − 2, n ≥ 2, and, as above, let H be a v-generic polarization. We recall the fundamental result on the second cohomology of the moduli space MH (v). Write v ⊥ ⊂ H ∗ (X, Z) for the sublattice orthogonal to v. Mukai introduced the natural homomorphism (2.18)

θv : v ⊥ → H 2 (MH (v), Z)  1 ∗ πM,∗ v(E ∨ ) · πX (x) 2 θv (x) 7→ ρ

where E is a quasi-universal family of similitude ρ (see [Mu3, Mu4]). Theorem 2.16 ([Hu2, OG1, Y1, Y2]). (1) The moduli space MH (v) is an irreducible pojective holomorphic symplectic manifold deformation equivalent to the Hilbert scheme of n points on X. (2) The homomorphism (2.18) is a Hodge isometry between v ⊥ and H 2 (MH (v), Z), where the lattice structure on the latter is given by the Beauville-Bogomolov form. Proof of Theorem 2.2 part 2. We treat first the case where an untwisted universal sheaf U exists over X × MH (v). Let η : O∆X → A = U ∨ [2] ◦ U be the morphism in D b (X × X) corresponding to the unit of the adjunction ΦU ⊣ ΨU (see [C2], Prop. 5.1). It suffices to show that pre-composition induces a surjection: ◦η

HomX×X (U ∨ [2] ◦ U , O∆X ) −→ HomX×X (O∆X , O∆X ). Note that since S∆X = O∆X [2], by (2.17), we may interpret this as a map Hom(U ∨ [2] ◦ U , S∆X [−2]) → HH−2 (X)∨ . Thus the construction (2.16) gives homomorphisms HH−2 (MH (v))∨ = Hom(O∆M

H (v)

, S∆M

◦η

H (v)

[−2]) → Hom(U ∨ [2] ◦ U , S∆X [−2]) → HH−2 (X)∨

whose transpose is the natural map in Hochschild homology induced by ΦU . So it is enough to show that ΦU ,∗ : HH−2 (X) → HH−2 (MH (v)) is injective. Now, note that Ie∗X (HH−2 (X)) ⊂

22

E. MARKMAN AND S. MEHROTRA

HΩ−2 (X) = H 2 (OX ) as Ie∗X is a graded map. Therefore, by Theorem 2.15, ΦU ,∗ |HH−2 (X) = (Ie∗M )−1 ◦ [ϕU ]2 ◦ Ie∗X ,

where [ϕU ]2 is the degree 2 part of the map on singular cohomology induced by ΦU . But observe that by the formula (2.18), θv = −[ϕU ]2 , whence, by Theorem 2.16, [ϕU ]2 is injective. This proves the result for fine moduli spaces. We sketch next the proof in the case where the universal sheaf U is twisted with respect to 2 (M (v), O ∗ ). Let F be an H-stable sheaf of class v and denote by [F ] a Brauer class α ∈ Han H ˆ → MH (v) be the blow-up centered at [F ]. The the corresponding point of MH (v). Let β : M 1 ∨ sheaf cohomology H (ΦU (F )) is an α-twisted reflexive sheaf of rank 2n − 2 over MH (v), which is locally free away from the point [F ], and the quotient V of β ∗ H 1 (ΦU (F ∨ )) by its ˆ be torsion subsheaf is a locally free β ∗ α-twisted sheaf [Ma5, Prop. 4.5]. Let p : P(V ) → M f := (1 × βp)∗ U , and let π the associated projective bundle. Set U ˜ij be the projection from X × PV × X to the product of the i-th and j-th factors. Set h i ∗ f∨ ∗ f ˜12 (U ) ⊗ π ˜23 U ⊗π ˜1∗ ωX [2] B := R˜ π13∗ π ∗ U ∨ ⊗ π ∗ U ⊗ π ∗ ω [2]] ∼ B, where Then we have the natural isomorphism A := Rπ13∗ [π12 = 1 X 23 the latter isomorphism follows from the projection formula and the isomorphism R(βp)∗ OPV ∼ = OMH (v) . The Brauer class p∗ β ∗ α is trivial, by [Ma2, Lemma 29(4)]. We thus have an equivalence of f is represented triangulated categories D b (X × PV, p∗ β ∗ α) ∼ = D b (X × PV ) and the image of U by an untwisted coherent sheaf G . We get an induced isomorphism  ∗ ∨  ∗ B∼ π13∗ π ˜12 G ⊗ π ˜23 G ⊗π ˜1∗ ωX [2] . = R˜

Replacing MH (v) by PV and E by G in Equation (2.18) we get an analogue θ˜v : v ⊥ → H 2 (PV, Z) of the Mukai homomorphism. The homomorphism θ˜v is the composition of p∗ β ∗ : H 2 (MH (v), Z) → H 2 (PV, Z) with the Mukai homomorphism (2.18). Indeed, the argument which shows that the Mukai homomorphism is independent of the choice of a quasi-universal sheaf shows also that using either (1×βp)∗ E or the direct sum G ⊕ρ of ρ copies of G in Equation (2.18) results in the same homomorphism. Theorem 2.15 applies now to the integral transform ΦG : D b (X) → D b (PV ) with kernel G and the argument in the case of untwisted universal sheaf goes through to show that O∆X is a direct summand of B, and hence of A as well.  3. Yoneda algebras

We prove Theorem 1.3 in this section. The reader interested only in the results on generalized deformations (Theorems 1.8 and 1.9) may skip this section. Let X be a projective K3 surface, M := MH (v) a moduli space of H-stable sheaves with Mukai vector v satisfying the hypothesis of Theorem 2.2, and ΦU : D b (X) → D b (M, θ) the faithful functor in Theorem 2.2. Assume that (v, v) ≥ 2. Definition 3.1. We say that the monad object A given in Equation (1.2) is totally split, if the composition α given in Equation (2.5) is an isomorphism. Assume for the rest of section 3 that the monad object A is totally split. This is the case if M is the Hilbert scheme X [n] and U is the universal ideal sheaf, by Theoem 2.2. For a more general sufficient condition for α to be an isomorphism, see Lemmas 6.9 and 6.16.

INTEGRAL TRANSFORMS AND DEFORMATIONS

23

Set pt := Spec(C) and let c : M → pt be the constant morphism. We get the object Y (OM ) := Rc∗ OM in D b (pt). As a graded vector space Y (OM ) is ⊕ni=0 H 2i (M, OM )[−2i], where n = dimC (M )/2. Given a graded vector space V , let 1Db (X) ⊗C V be the endofunctor of D b (X) sending an object x to x ⊗C V . Set Υ := 1Db (X) ⊗C Y (OM ),

R := 1Db (X) ⊗C H 2n (M, OM )[−2n].

We define next a natural transformation Write h =

Pn

i=0 h2i

h : R → Υ. according to the direct sum decomposition of Y (OM ), so that

h2i : 1Db (X) ⊗C H 2n (M, OM )[−2n]



1Db (X) ⊗C H 2n−2i (M, OM )[2i − 2n].

Let tX be a non-zero element of H 2 (X, OX ), considered as the subspace H 0,2 (X) of the complexified Mukai lattice, and let tM be its image in H 2 (M, OM ) via Mukai’s Hodge isometry j−1 (2.18). Denote by t∗M : Y (OM ) → Y (OM ) the homomorphism, which sends tjM to tM , 1 ≤ j ≤ n, and sends 1 to 0. The choice of tM identifies h2i as an element of the Hochschild ˜ 2i ⊗(t∗ )i , where h ˜ 2i belongs to Ext2i (O∆ , O∆ ). cohomology HH 2i (X). Explicitly, h2i = h X×X X X M Let σX be the class in H 0 (X, ωX ) dual to the class tX with respect to Serre’s duality. Given h2 , ˜ 2 ⊗ σX in HomX×X (∆X,∗ (OX ), ∆X,∗ (ωX )[2]) is a class in HH0 (X), which depends the class h canonically on h2 and is independent of the choice of the class tX , since tM depends on tX linearly. Hence, the choice of h2 corresponds to a choice of a class in HH0 (X), which we denote by h2 as well. Let h2 be the class in HH0 (X) which is mapped to the Chern character ch(v) in HΩ0 (X) of sheaves with Mukai vector v via the Hochschild-Kostant-Rosenberg isomorphism I∗X : HH0 (X) → HΩ0 (X). I∗X (h2 ) := ch(v). Set h2i := (−1)i+1 (h2 )i . Explicitly, (3.1)

h0 = −1, ˜ 2 )2 ⊗ (t∗ )2 , h4 = −(h M hk = 0, for k > 4.

Given an object x in D b (X), let hx : x ⊗C H 2n (M, OM )[−2n] → x ⊗C Y (OM ) be the morphism induced by the natural transformation h. Let πX : X × M → X be the ∗ . Denote4 by projection. Note that the endo-functor Υ is naturally isomorphic to RπX∗ ◦ πX D b (X)T the full subcategory of D b (M ) with objects of the form ΦU (x), for some object x in D b (X). Let ΞU : D b (X × M ) → D b (M ) be the composition of tensorization by U followed ∗ . Denote by D b (X) the full subcategory of D b (X × M ) with by RπM∗ . Then ΦU = ΞU ◦ πX e Y ∗ (x), for some object x in D b (X). Let Q : D b (X) → D b (X) be the objects of the form πX T e Y restriction of the functor ΞU . Theorem 3.2. The natural transformation q : Υ → ΨU ΦU , given in Equation (2.4) above, has the following properties. 4Db (X)

T is the Kleisli category associated to the adjunction ΦU ⊣ ΨU . The subscript T will later denote the monad associated to this adjunction, so that our notation is the standard one [Mac, Sec. VI.5].

24

E. MARKMAN AND S. MEHROTRA

(1) For every pair of objects x1 and x2 in D b (X), the first row below is a short exact sequence. 0

/ Hom(x1 , Rx2 )

(hx2 )∗

/ Hom(x1 , Υx2 )

(qx2 )∗

/ Hom(x1 , ΨU ΦU (x2 )) → 0 ∼ =

∼ =



∗ (x ), π ∗ (x )) HomDb (X×M ) (πX 1 X 2

Q

 / Hom b (Φ U (x1 ), ΦU (x2 )). D (M )

(2) The vertical adjunction isomorphisms above conjugate (qx2 )∗ to the homomorphism induced by the functor Q : D b (X)Ye → D b (X)T . The functor Q is full.

The theorem is proven in section 3.4. Given objects x and y in a bounded triangulated category, set Hom• (x, y) := ⊕k∈Z Hom(x, y[k])[−k]. Note that Hom•Db (X) (x1 , Υx2 ) is simply Hom• (x1 , x2 ) ⊗C Y (OM ). An explicit calculation of the algebra Hom• (ΦU (x), ΦU (x)) as a quotient of Hom• (x, x) ⊗C Y (OM ) is carried out in Theorem 3.21 for any object x represented by a simple sheaf. Section 3 is organized as follows. In subsection 3.1 we interpret Theorem 3.2 in terms of the standard construction of a quotient category by a congruence relation (Definition 1.2). The natural transformation h gives rise to such a relation and Theorem 3.2 expresses the full subcategory D b (X)T of D b (M ) as the quotient category of the category D b (X)Ye , whose objects are the same as those of D b (X), and such that Hom•Db (X)e (x1 , x2 ) = Hom•Db (X) (x1 , x2 ) ⊗C C[t]/(tn+1 ), Y

where t has degree 2. In subsection 3.2 we show that the natural transformation q : Υ → T in Theorem 3.2 induces a monad map between two monads in D b (X). Under the analogy between rings and monads, the statement that q is a monad map says that q is analogous to a ring homomorphism. The functor Q in Theorem 3.2 is an example of the general construction of a Kleisli lifting of a monad map [MaMu, Theorem 2.2.2 and Def. 2.2.3]. In subsection 3.3 the natural transformation h is defined formally in terms of a cone of q. In subsection 3.4 we reduce the proof of Theorem 3.2 to the computation of the component h2 of the natural transformation h. Subsections 3.5 to 3.7 are dedicated to the proof that h2 is the inverse image of ch(v) via the Hochschild-Kostant-Rosenberg isomorphism. The class tX in H 2 (X, OX ) gives rise to a natural transformation 1Db (X) → 1Db (X) [2] and hence a morphism x → x[2], for every object x of D b (X). We get two morphisms from ΦU (x) to ΦU (x)[2] in D b (M ), one is the image of the former via the functor ΦU . The second is induced by the natural transformation 1Db (M ) → 1Db (M ) [2] associated to the image tM in H 2 (M, OM ) of tX via Mukai’s Hodge isometry. These two morphisms in HomDb (M ) (ΦU (x), ΦU (x)[2]) are linearly independent in general, but their traces belong to the one-dimensional space H 2 (M, OM ). In subsection 3.5 we calculate the ratio of the two traces using Hochschild cohomology techniques. When x is a simple object, the compositions of each of the two morphisms in HomDb (M ) (ΦU (x), ΦU (x)[2]) n−1 are linearly dependent in the one-dimensional space HomDb (M ) (ΦU (x), ΦU (x)[2n]). with tM In subsection 3.6 we relate the ratio of the latter pair to the ratio of traces computed earlier. We get a relation in the Yoneda algebra of ΦU (x), for every simple object x. In subsection 3.7 we use that relation to determine the component h2 of the natural transformation h and complete the proof of Theorem 3.2. In subsection 3.8 we relate moduli spaces of stable sheaves on X to certain moduli spaces of sheaves over M .

INTEGRAL TRANSFORMS AND DEFORMATIONS

25

3.1. A congruence relation associated to the natural transformation h. We describe in this subsection how Theorem 3.2 reconstructs the category D b (X)T as a quotient category in the sense of Definition 1.2. Let D b (X)Ye be the category whose objects are the same as those of D b (X), and such that HomDb (X)e (x1 , x2 ) := HomDb (X) (x1 , Υx2 ) = Y

2n M k=0

HomDb (X) (x1 , x2 [−2k]) ⊗C H 2k (M, OM ).

Note that Hom•Db (X) (x1 , x2 ) = Hom•Db (X) (x1 , x2 )⊗C Y (OM ). Given morphisms g in HomDb (X) (x1 , x2 ) Ye and f in HomDb (X) (x2 , x3 ), and elements a, b in Y (OM ), the composition (f ⊗ a) ◦ (g ⊗ b) is f g ⊗ ab, and is extended by linearity to all morphisms. Note that D b (X)Ye is equivalent ∗ (x), for some obto the full subcategory of D b (X × M ) whose objects are of the form πX ject x in D b (X). The above composition rule corresponds to composition in D b (X × M ) ∗ x , π ∗ x ). Let π ∗ : via the adjunction isomorphism HomDb (X) (x1 , Υx2 ) ∼ = HomDb (X×M ) (πX 1 X 2 X b b D (X) → D (X)Ye be the functor sending each object to itself and inducing the natural inclusion HomDb (X) (x1 , x2 ) → HomDb (X)e (x1 , x2 ). Y Definition 1.2 recalls the notion of a congruence relation on a category. Consider the relation R on D b (X)Ye given in Equation (1.4). Following is a restatement of Theorem 3.2 in the language of quotient categories. Theorem 3.3. (1) R is a congruence relation. (2) The natural transformation q induces a fully faithful functor Σ : D b (X)Ye /R → D b (M ). (3) The functor ΦU factors through the quotient functor Q : D b (X)Ye → D b (X)Ye /R as the ∗ : D b (X) → D b (M ). composition ΦU = ΣQπX Proof. The statement follows from Theorem 3.2. It is however instructive to see how the defining Equations (3.1) of h formally imply that R is a congruence relation, so we will prove part 1 of Theorem 3.3 independently. The image of (hx2 )∗ , given in (1.4), is mapped under post-composition with elements g of ∗ [Hom πX D b (X) (x2 , x3 )] to the image of (hx3 )∗ : HomD b (X) (x1 , Rx3 ) → Hom(x1 , Υx3 ), by the ∗ (˜ naturality of the transformation h. Indeed, if g = πX g ) and f belongs to HomDb (X) (x1 , Υx2 ), then g ◦ f = Υ(˜ g ) ◦ f and naturality of h yields the following commutative diagram: HomDb (X) (x1 , Rx2 )

(hx2 )∗

/ Hom b D (X) (x1 , Υx2 ) Υ(˜ g )∗

R(˜ g )∗



HomDb (X) (x1 , Rx3 )

 / Hom b D (X) (x1 , Υx3 ).

(hx3 )∗

The algebra Y (OM ) is generated by the element tM . Let τ : Υ → Υ[2] be the natural transformation corresponding to multiplication by tM . We may regard the ˜2 : 1 b ˜ natural transformation h D (X) → 1D b (X) [2] as a natural transformation h2 : R → R[2] as well. Then we have the equality ˜ 2, τ ◦ h = −h ◦ h

26

E. MARKMAN AND S. MEHROTRA

which follows from the defining Equations (3.1) of h. Indeed,    0 0   ..  ..    . .       0 0 h= , τ ◦ h =  ˜ 2 ⊗ (t∗ )2    −h 0 2 M    ˜ 2 ⊗ t∗ ∗ ˜2 ⊗ t   −h  h 2 M M ˜2 h −1



    ˜2.  = −h ◦ h   

It follows that the image of (hx2 )∗ , given in (1.4), is mapped under post-composition with 1⊗tM ∈ HomDb (X)e (x2 , x2 [2]) to the image of (hx2 [2] )∗ : HomDb (X) (x1 , Rx2 [2]) → Hom(x1 , Υx2 [2]). Y The analogous statement holds for powers of 1 ⊗ tM , by induction. Hence, post-composition with every element g ∈ HomDb (X)e (x2 , x3 ) maps the image of (hx2 )∗ to the image of (hx3 )∗ . Y Let e be an element of HomDb (X)e (x0 , x1 ). We need to show that pre-composition with e Y maps the image of (hx2 )∗ : HomDb (X) (x1 , Rx2 ) → HomDb (X) (x1 , Υx2 ) to the image of (hx2 )∗ : ∗ (˜ HomDb (X) (x0 , Rx2 ) → HomDb (X) (x0 , Υx2 ). If e = πX e), where e˜ belongs to HomDb (X) (x0 , x1 ), then for a ∈ HomDb (X) (x1 , Rx2 ) we have (hx2 ◦ a) ◦ e = hx2 ◦ (a ◦ e˜).

The right hand side above belongs to the image of (hx2 )∗ . It remains to prove the statement ∗ (t ) : 1 in case x0 = x1 [−2] and e = 1 ⊗ tM . Now πM M D b (X×M ) → 1D b (X×M ) [2] is a natural transformation. It follows that for every pair of objects y1 , y2 in D b (X × M ) and for every morphism f : y1 → y2 we have the commutative diagram y1

f

/ y2

∗ (t ) πM M

∗ (t ) πM M



y1 [2]



[2](f )

/ y2 [2].

∗ (x ) and for f := h This holds in particular for objects yi of the form πX i x2 ◦ a. We get the equality (hx2 ◦ a) ◦ (1 ⊗ tM ) = (1 ⊗ tM ) ◦ [2](hx2 ◦ a), for all a ∈ HomDb (X) (x1 , Rx2 ). Post-compositions were shown already to preserve the relation R. 

3.2. The monad A is a quotient of a constant monad. Set T := ΨU ΦU : D b (X) → D b (X).

Denote by η : 1 → T the unit for the adjunction ΦU ⊣ ΨU , by ǫ : ΦU ΨU → 1 the counit, and set m := ΨU ǫΦU : T 2 → T the multiplication natural transformation. We get the monad

(3.2)

T := (T, η, m).

Let A be the object in D b (X × X) given in Equation (1.2). Recall that A is the kernel of the integral transform T . We have the natural morphism m : A ⊗C Y (OM ) → A ,

given in Equation (2.3). Let η : O∆X → A be the morphism corresponding to the unit of the adjunction ΨU ⊣ ΦU . Set (3.3)

q := m ◦ (η ⊗ id) : O∆X ⊗C Y (OM ) → A .

INTEGRAL TRANSFORMS AND DEFORMATIONS

27

The Yoneda algebra Y (OM ), considered as an object of D b (pt), is a monad in an obvious way. Denote by η˜ : 1 → Y (OM ) its unit and by m ˜ : Y (OM ) ⊗C Y (OM ) → Y (OM ) its multiplication. The endo-functor Y (OM ) ⊗C (•) : D b (X) → D b (X), of tensorization by Y (OM ) over C, has kernel Y := O∆X ⊗C Y (OM ). We get a monad e := (Υ, η˜, m) Y ˜

(3.4)

in D b (X) as follows. Denote by πX : X × M → X the projection. We have the adjunction ∗ ⊣ Rπ , Y is the kernel of the functor Υ := Rπ ∗ e πX X∗ X∗ ◦ πX , and Y is the monad for that adjoint pair. We denote again by q the natural transformation from Υ to T induced by the homomorphism of kernels (3.3). Remark 3.4. The above definition of q is the one we used earlier in Equation (2.4). The natural transformation q admits a second functorial expression, which will be needed below (in Lemma 3.7). Let ΞU : D b (X × M ) → D b (M ) be the composition of tensorization by ∗ . Denote by G the right adjoint of Ξ . Set U followed by RπM∗ . Then ΦU = ΞU ◦ πX U ≈ ∗ ∗ F := RπX∗ , so that Υ = F πX and T = F GΞU πX . Let η : 1Db (X×M ) → GΞU be the unit for ≈

∗ . Similarly, the adjunction. Then the natural transformation q : Υ → T is equal to F η πX the homomorphism q in (3.3) admits an analogous description provided below. Let ∆2,3 : X × M → (X × M ) × M be the diagonal map. The kernel of the integral ∗ ω [2]). The kernel K of GΞ functor ΞU is ∆2,3∗ (U ). The kernel of G is ∆2,3∗ (U ∨ ⊗ πX X U is b their convolution and is identified as an object in D ((X × M ) × (X × M )) as follows. Let ∆2,4 : X × M × X → (X × M ) × (X × M ) be the diagonal map. Let πi,j be the projection from X × M × X onto the product of the i-th and j-th factors. Then  ∗ ∗ ∗ K = ∆2,4∗ π1,2 (U ∨ ⊗ πX ωX [2]) ⊗ π2,3 U .

Let pi1 ···ik be the projection from X × M × X × M onto the product of the i1 , . . . , ik factors. ≈ Let η : ∆X×M∗ OX×M → K be the unit morphism for the adjunction ΞU ⊣ G. Then Rp13∗ : D b ((X × M ) × (X × M )) → D b (X × X) ≈

maps ∆X×M∗ OX×M to the object Y , maps K to the object A , and maps the morphism η to the morphism q given in Equation (3.3). The latter equality is proven in the following Lemma. ≈

Lemma 3.5. The equality q = Rp13∗ ( η ) holds. ≈

∗ Rπ Proof. Set q ′ := Rp13∗ (η ). Let γ : π13 13∗ → 1D b (X×M ×X) be the counit for the adjunction ∗ ∗ be the unit natural transformation for this π13 ⊣ Rπ13∗ . Let u : 1Db (X×X) → Rπ13∗ π13 ∗ adjunction. Let η˜ : O∆X → Rπ13∗ π13 O∆X be the morphism associated by u to the object ∗ ⊣ Rπ . Set O∆X . The morphism η˜ is itself the unit morphism for the adjunction πX X∗ Af:= Rp123∗ (K ), so that Rπ13∗ Af∼ = A . We have   ∗ ∗ q := m ◦ (η ⊗ id) = Rπ13∗ (γAe) ◦ (Rπ13∗ π13 (η)) = Rπ13∗ γAe ◦ π13 η .  ≈  ≈ On the other hand, q ′ = Rp13∗ (η ) = Rπ13∗ Rp123∗ (η ) . Hence, it suffices to prove the equality

(3.5)



∗ η. Rp123∗ (η ) = γAe ◦ π13

28

E. MARKMAN AND S. MEHROTRA ≈

We have the equality η = Rp13∗ ( η ) ◦ η˜, since η is the unit morphism for the adjunction ∗ . We get ΦU ⊣ ΨU , where ΦU = ΞU ◦ πX  ≈  ∗ ∗ ∗ (˜ η ), π13 η = π13 Rπ13∗ Rp123∗ ( η ) ◦ π13 and Equation (3.5) becomes  ≈  ≈ ∗ ∗ (˜ η ). Rπ13∗ Rp123∗ (η ) ◦ π13 Rp123∗ ( η ) = γAe ◦ π13



∗ , G = Rπ The latter is a special case of Lemma 3.6 applied with F = π13 13∗ , f = Rp123∗ ( η ), A = O∆X , and B = Af. 

Let F : C → D be a functor, F ⊣ G an adjunction, u : 1C → GF the unit, and γ : F G → 1D the counit for the adjunction. Let A be an object of C , let B be an object of D, and let f : F (A) → B be a morphism. Lemma 3.6. f = γB ◦ F G(f ) ◦ F (uA ). γF (A)

F (uA )

Proof. The composition F (A) → F ((GF )(A)) = ((F G)F )(A) → F (A) is the identity, by [Mac, Theorem 1]. Hence, it suffices to prove the equality γB ◦ F G(f ) = f ◦ γF (A) . The latter equality follows from the commutativity of the following diagram. Set A′ := F (A). F / G/ Hom(G(A′ ), G(B)) Hom(F G(A′ ), F G(B)) ❘❘❘ ❘❘❘ ✐✐✐✐ ❘❘❘ ✐✐✐✐ ∼ ✐ adj ✐ = ✐ ❘ ✐ ∗ ❘❘❘ γA ′ ) t✐✐✐✐ (γB )∗ 

Hom(A′ , B)

Hom(F G(A′ ), B).

The left triangle above commutes, by [Hu1, Lemma 1.21]. The proof of the commutativity of the right triangle is similar.  Lemma 3.7. The natural transformation q is a monad map5 in the sense that q η˜ = η and the following diagram commutes: Y9 ◦ A

qA

r

Y q rrrr

rr rrr Y ◦ Y▲ ▲▲▲ ▲▲▲ ▲▲▲ m ˜ ▲%

/ A ◦A m

Y

q

 /A.

Proof. The equality q η˜ = η is clear. We prove only the commutativity of the above diagram. Let B, C, D be categories, let G : B → C and F : C → D be functors. Assume given adjunctions G∗ ⊣ G and F ∗ ⊣ F . Let η and ǫ be the unit and counit for G∗ ⊣ G. Define η˜, ǫ˜ similarly for F ∗ ⊣ F , and let m ˜ := F ˜ ǫF ∗ be the multiplication for the corresponding monad. Set Ψ := F G, Φ := G∗ F ∗ , T := ΨΦ = F GG∗ F ∗ , and Y := F F ∗ . Let the natural transformation q : Y → T 5We use the the term monad map following [MaMu, Def. 2.2.3]. A monad map is a special case of a monad

functor between two monads in different categories [St].

INTEGRAL TRANSFORMS AND DEFORMATIONS

29

be given by q := F ηF ∗ : F F ∗ → F GG∗ F ∗ . We claim that q is a monad map, in the sense that the following diagram commutes YT ②< Y q ②②② ②② ②②

qT

Y Y❋

/ TT

m

❋❋ ❋❋ ❋❋ ❋❋ m ˜ "

Y

q

 / T.

The above diagram is obtained by applying F on the left and F ∗ on the right to the circumference of the following diagram F ∗9 F GG∗ r

F ∗ F ηrrrr

rr rrr

F ∗ F❑

ηF ∗ F GG∗ ∗ ∗ / GG F F GG∗ GG∗ ˜ ǫGG∗

ǫ˜GG∗





GG O ❑❑ ❑❑❑ η ❑❑❑ ǫ ˜ ❑❑ %

ηGG∗

1C



/ GG∗ GG∗ GǫG∗



/ GG∗ .

η

The left triangle and the two right squares in the above diagram evidently commute. Apply the above argument with F := RπX∗ : D b (X × M ) → D b (X) and with the functor ∗ with tensorization by the object G : D b (M ) → D b (X × M ) given by the composition of πM ∨ b ∗ →T U [2] in D (X × M ). (G is the right adjoint of ΞU ). This establishes that q : RπX∗ πX is a monad map. Every step in the above argument admits an evident translation to the case of integral functors. Note that we used above the description of q given in Lemma 3.5.  Remark 3.8. The monad map q, induced by the morphism q : Y → A of Fourier-Mukai kernels, induces a functor e

P : D b (X)T → D b (X)Y

e := (Υ, η˜, m) between the categories of modules for the monads T := (T, η, m) and Y ˜ in D b (X) [J, Lemma 1]. The functor P takes the T-module (x, a) to P (x, a) = (x, a ◦ qx )



e

D b (X)Y .

The functor P is faithful, as the homomorphisms spaces are both subspaces of those of D b (X). The functor P is an example of an Eilenberg-Moore lifting of a monad functor, where the monad functor in our case is (1Db (X) , q) [MaMu, Def. 2.2.1]. Under the analogy between the monad map q and an algebra homomorphism, the functor P corresponds to the change of scalars functor, or to push-forward. The functor Q in Theorem 3.2 is analogous to a pull-back functor and goes in the opposite direction. For that reason the functor P will not play a role below. 3.3. A universal relation “ideal”. The following proposition introduces a “universal relation ideal” R. Consider the object R := O∆X [−2n] ⊗C Ext2n (OM , OM ) in D b (X × X). Proposition 3.9. Assume that the monad A is totally split (Definition 3.1). There exists a morphism h : R → Y , unique up to a scalar factor, such that the following is an exact

30

E. MARKMAN AND S. MEHROTRA

triangle, which admits a splitting. h

(3.6)

q

0

R −→ Y −→ A → R[1].

Proof. There exists an object R ′ in D b (X × X) and a morphism h : R ′ → Y such that h

q

R ′ → Y → A → R ′ [1] is an exact triangle, by the axioms of a triangulated category. The following composition α ι

q

O∆X ⊗C λn −→ Y −→ A , given in Equation (2.5), is an isomorphism by the assumption that the monad is totally split. Using the long exact sequence in sheaf cohomology coming from the exact triangle q h R ′ → Y → A , one immediately obtains R ′ ∼ = O∆X [−2n] ⊗C Ext2n (OM , OM ). The triangle is split as there are no odd-degree self-extensions of O∆X . Finally, h is determined up to scalars  as the automorphism group of the object R ′ ∼ = O∆X [−2n] is C∗ . The morphism h : R → O∆X ⊗ Y (OM ) is naturally an element of (3.7)

 ⊕2j=0 Ext2j (O∆X , O∆X ) ⊗C Hom H 2n (M, OM ), H 2n−2j (M, OM ) .

Let tX be a non-zero element of H 2 (X, OX ), considered as a subspace of the complexified Mukai lattice, and let tM be its image in H 2 (M, OM ) via Mukai’s Hodge isometry (2.18). j−1 Denote by t∗M : Y (OM ) → Y (OM ) the homomorphism, which sends tjM to tM , 1 ≤ j ≤ n, and sends 1 to 0. The choice of tM identifies h as an element of the Hochschild cohomology ˜ 0 ⊗1+ h ˜ 2 ⊗t∗ + h ˜ 4 ⊗(t∗ )2 , where h ˜ 2j belongs to Ext2j (O∆ , O∆ ). HH ∗ (X). Explicitly, h = h X X M M Let σX be the class in H 0 (X, ωX ) dual to the class tX with respect to Serre’s duality. The ˜ 2 ⊗ σX in HomX×X (∆X,∗ (OX ), ∆X,∗ (ωX )[2]) is a class in HH0 (X), independent class h2 := h of the choice of the class tX , since tM depends on tX linearly. ˜ 0 does not vanish and h ˜0h ˜ 4 = (h ˜ 2 )2 . Theorem 3.10. (1) The class h ˜ 0 = −1. Then the class I X (h2 ) in HΩ0 (X) is equal (2) Rescale the morphism h, so that h ∗ to the Chern character ch(v) of the Mukai vector v of sheaves parametrized by M . Proof. Part 2 of the theorem is proven in section 3.7. We include here the proof of part ˜ 0 does not vanish, since the sheaf 1, which follows formally from Lemma 3.7. The class h 2n 2n cohomology H (Y ) does not vanish, while H (A ) vanishes. It remains to compute ˜h4 . Identify A with ∆∗ OX ⊗C λn via the isomorphism α given in Equation (2.5). The morphism q : Y → A decomposes q = (qi,j ), 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n, where qi,j is a morphism qi,j : ∆∗ OX ⊗C H 2j (M, OM )[−2j] → ∆∗ OX ⊗C H 2i (M, OM )[−2i]. Then qi,i is the identity, for 0 ≤ i ≤ n − 1, and qi,j = 0 for i 6= j and 0 ≤ j ≤ n − 1, by construction of α. Note that we use here the equality of the two descriptions of q in Lemma 3.5, as α was constructed in terms of the earlier description, while Lemma 3.7, soon to be applied, uses the second description. The morphism h : R → Y decomposes as a column h = (hi,n ), where n is fixed, 0 ≤ i ≤ n, and hi,n is a morphism hi,n : ∆∗ OX ⊗C H 2n (M, OM )[−2n] → ∆∗ OX ⊗C H 2i (M, OM )[−2i].

INTEGRAL TRANSFORMS AND DEFORMATIONS



0 if 0 ≤ i ≤ n − 3 ˜ 2(n−i) ⊗ (t∗ )n−i if n − 2 ≤ i ≤ n. h M  0    ..  . q0,n     . 0 .. (qi,j ) =  Iλ h= , n  hn−2,n  qn−1,n  hn−1,n hn,n

Clearly, hi,n =

We have the equality 

q0,n hn,n .. .

   0 = qh =  qn−3,n hn,n   qn−2,n−2 hn−2,n + qn−2,n hn,n qn−1,n−1 hn−1,n + qn−1,n hn,n

We get the equalities:









31

0 .. .

        0 = ˜   h4 ⊗ (t∗M )2   ˜ 2 ⊗ t∗   h M ˜0 h ˜0 q0,n h .. . ˜0 qn−3,n h



    .   

      =   ˜ 4 ⊗ (t∗ )2 + qn−2,n h ˜0   h M ˜ 2 ⊗ t∗ + qn−1,n h ˜0 h M



   .  

(3.8)

qi,n = 0, for 0 ≤ i ≤ n − 3, ˜4 h (3.9) qn−2,n = − ⊗ (t∗M )2 , ˜0 h ˜2 h qn−1,n = − ⊗ t∗M . (3.10) ˜0 h The class tM yields a morphism t : ∆∗ OX [−2] → Y , which is an embedding of ∆∗ OX [−2] as a direct summand of Y . We get the commutative diagram Y [−2]

q

/ A [−2] tA

tY





Y ◦Y

Yq

/ Y ◦A.

The morphism Y q in the above diagram appears also in the commutative diagram in the statement of Lemma 3.7. Glue the two diagrams along the arrow Yq . Set τ := m(qA )(tA ) : A [−2] → A and τ˜ := m(tY ˜ ) : Y [−2] → Y . The commutativity of these two diagrams yields the equality τ q = q˜ τ : Y [−2] → A .   ~0 0 , where In is the n × n identity matrix and The matrix of τ˜ is (˜ τi,j ) = In ⊗ tM ~0 tM : H 2j (M, OM )f →H 2j+2 (M, OM ) is the isomorphism obtained by multiplication by the class tM , for 0 ≤ j ≤ n − 1. Hence,     0 0 ..   ..   .     .     0 ,  τ˜h =  τ h) =   and we get 0 = τ (qh) = q(˜ 0  ˜ ˜   h h ∗ 2 4 2   − ˜h ⊗ (tM ) ˜ 4 ⊗ t∗   h   M 0 ˜ 2 )2 (h ˜2 ∗ ∗ ˜ h h4 ⊗ tM − h˜ ⊗ tM 0

32

E. MARKMAN AND S. MEHROTRA

˜ 0h ˜4 = where the last equality follows from equations (3.8), (3.9), and (3.10). The equality h 2 ˜ 2 ) follows. (h  3.4. Computation of the full subcategory D b (X)T of D b (M ). Let D b (X)Ye be the full ∗ (x), for some object x in D b (X). subcategory of D b (X ×M ) consisting of objects of the form πX e : D b (X)e → D b (X × M ). Let ΞU : D b (X × M ) → We get a natural full and faithful functor Σ Y ∗ . D b (M ) be the composition of tensorization by U followed by RπM∗ . Then ΦU = ΞU ◦ πX We get the following commutative diagram, where the functor Q is the restriction of ΞU . D b (X)Ye

(3.11)

: tt tt t tt ∗ tt πX

D b (X)

e Σ

ΞU

Q

❏❏ ❏❏ ΦU ❏❏ ❏❏ ❏%

/ D b (X × M )



D b (X)T

Σ

 / D b (M )

Let qxi : Υ(xi ) → T (xi ) be the morphism induced by the natural transformation q, which in turn is induced by the morphism of kernels given in Equation (3.3). We get the homomorphism (qx2 )∗ : HomDb (X) (x1 , Υx2 ) → HomDb (X) (x1 , T x2 ) f

and the diagram:

7→ qx2 ◦ f, ∼ =

∗ (x ), π ∗ (x )) HomDb (X×M ) (πX 1 X 2

(3.12)

/ Hom b D (X) (x1 , Υx2 ) (qx2 )∗

ΞU



HomDb (X) (ΦU (x1 ), ΦU (x2 ))

∼ =



/ Hom b D (X) (x1 , T x2 ).

where the horizontal isomorphisms are due to the adjunctions. Lemma 3.11. The above diagram is commutative. Proof. The commutativity is a special case of the following Lemma applied with F := RπX∗ , G∗ := ΞU , B := D b (M ), C := D b (X × M ), and D := D b (X).  Let B, C , and D be categories, let G : B → C and F : C → D be functors, and let ⊣ G and F ∗ ⊣ F be adjunctions. Let η : 1C → GG∗ be the unit for the adjunction. Set q := F ηF ∗ : F F ∗ → F GG∗ F ∗ . G∗

Lemma 3.12. The following diagram is commutative for every pair of objects x1 , x2 in D. ∼ =

Hom(F ∗ x1 , F ∗ x2 ) o

G∗

❱❱❱❱ ❱❱❱(η❱F ∗ x2 )∗ ❱❱❱❱ ❱❱❱❱ ❱*

Hom(x1 , F F ∗ x2 )

Hom(F ∗ x1 , GG∗ F ∗ x2 )

✐✐✐✐ ✐✐✐✐ ✐ ✐ ✐ ✐✐ ∼ = t✐✐✐✐  ∗ ∗ ∗ ∗ Hom(G F x1 , G F x2 ) o

❯❯❯❯ ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ ∼ = ❯*

∼ =



(qx2 )∗

Hom(x1 , F GG∗ F ∗ x2 ).

INTEGRAL TRANSFORMS AND DEFORMATIONS

33

Proof. All the arrows labeled as isomorphisms correspond to the adjunction isomorphisms. Hence, the lower middle triangle commutes. The left triangle commutes, by [Hu1, Lemma 1.21]. The upper right triangle commutes, by definition of q and the naturality of the adjunction isomorphisms.  Proof of Theorem 3.2. Theorem 3.10 yields the exact sequence with the natural transformation h satisfying Equations (3.1). The sequence in the statement of Theorem 3.2 is short exact, by the the splitting of the exact triangle (3.6). The diagram in Theorem 3.2 is commutative, by Lemma 3.11. The functor Q is full, by the surjectivity of (qx2 )∗ in the diagram in Theorem 3.2 and the commutativity of that diagram.  3.5. Traces. Subsections 3.5 to 3.7 are dedicated to the proof of part 2 of Theorem 3.10. Given two smooth projective varieties X and M , an integral functor Φ : D b (X) → D b (M ), and an object x ∈ D b (X), we have a natural composite homomorphism µx

Φ

tr

H i (X, OX ) −→ Hom(x, x[i]) −→ Hom(Φ(x), Φ(x)[i]) −→ H i (Y, OY ). The definition of the natural transformations µ and tr are recalled below. In this subsection we use known results about the functoriality of Hochschild homology in order to provide a topological formula for the homomorphism displayed above in a special case (see Proposition 3.16). The following lemma will be needed in the proof of Proposition 3.16 below. Let X and Y be smooth projective varieties, f : X → Y a morphism, x an object of D b (X) and y an object of D b (Y ). We will use the notation f∗ and f ∗ for the right and left derived functors Rf∗ and Lf ∗ for brevity. Assume given morphisms t : x → f ∗ y and φ : f ∗ y → x ⊗ ωX [dim X]. Let η : 1 → f∗ f ∗ be the unit for the adjunction f ∗ ⊣ f∗ . Let f! be the left adjoint of f ∗ and let ηy (t) ∈ Hom (f! (x), f∗ f ∗ (y)) be the image of t via the composition ∗

(3.13)

f (ηy ) Hom(x, f ∗ (y)) −→ Hom(x, f ∗ f∗ f ∗ (y)) ∼ = Hom(f! (x), f∗ f ∗ (y)).

Note that φ◦t belongs to Hom(x, x⊗ωX [dim X])). Using the isomorphism f∗ (x⊗ωX [dim X]) ∼ = f! (x) ⊗ ωY [dim Y ] we see that f∗ (φ) ◦ ηy (t) belongs to Hom(f! (x), f! (x) ⊗ ωY [dim Y ]). Let (3.14)

T rX : Hom(x, x ⊗ ωX [dim X])

−→ C be the composition of the isomorphism Hom(x, x⊗ωX [dim X]) ∼ = Hom(x, x)∗ induced by Serre duality, followed by evaluation Hom(x, x)∗ → C on the identity morphism in Hom(x, x). Lemma 3.13. The following equality holds. T rX (φ ◦ t)

=

T rY (f∗ (φ) ◦ ηy (t)).

Proof. Given z ∈ D b (Y ), we get the adjunction isomorphism (3.15)

∼ =

Hom(x, f ∗ z) −→ Hom(f! x, z).

Serre duality yields the dual isomorphism: ∼ =

Hom(f ∗ z, SX x) ←− Hom(z, SY f! x).

Thus given t′ ∈ Hom(x, f ∗ z) and φ′ ∈ Hom(f ∗ z, SX x), we have an equality (3.16)

T rX (φ′ ◦ t′ ) = T rY (φ′ ◦ ǫ ◦ f! (t′ ))

where φ′ ∈ Hom(z, SY f! x) is the preimage of φ′ , while ǫ(f! (t′ )) ∈ Hom(f! x, z) is the adjoint map to t′ , where ǫ : f! f ∗ → 1 is the counit.

34

E. MARKMAN AND S. MEHROTRA

Replace z by f∗ f ∗ y in (3.15), and pre-compose with the unit to get the diagram (3.13) whose Serre dual is (3.17) ∼ =

◦f ∗ η

Hom(f ∗ y, SX x) ←− Hom(f ∗ f∗ f ∗ y, SX x) ←− Hom(f∗ f ∗ y, SY f! x) = Hom(f∗ f ∗ y, f∗ SX x).

Given t ∈ Hom(x, f ∗ y), its image in Hom(f! x, f∗ f ∗ y) via (3.13) is nothing but ηy (t). Similarly, for φ ∈ Hom(f ∗ y, SX x), the preimage of φ ◦ ǫf ∗ y ∈ Hom(f ∗ f∗ f ∗ y, SX x) in Hom(f∗ f ∗ y, f∗ SX x) via the middle isomorphism in (3.17) is easily verified to be f∗ (φ). Thus, equation (3.16) implies that T rX ((φ ◦ ǫf ∗ y ) ◦ (f ∗ ηy ◦ t)) = T rY (f∗ (φ) ◦ ηy (t)). Finally, since ǫf ∗ y ◦ f ∗ ηy = idf ∗ y , we have that (φ ◦ ǫf ∗ y ) ◦ (f ∗ ηy ◦ t) = φ ◦ t, and the result follows.  Let ν be a class in HH0 (X) ∼ = Hom(∆X,∗ OX , ∆X,∗ ωX [dim X]) and let x be an object of Regarding ν as a natural transformation from 1Db (X) to the Serre functor we get a morphism νx : x → x ⊗ ωx [dim X]. Given a class c in HΩ∗ (X), denote by cp,q the direct summand in H p (X, ΩqX ). Set d := dim(X). h i  p ∨ X e . Lemma 3.14. The following equality holds: T rX (νx ) = T rX I∗ (ν) tdX ch(x ) D b (X).

d,d

Note that the right hand side is the Mukai pairing of ν and the class (Ie∗X )−1 (ch(x)) in HH0 (X) as defined in [C2, Def. 6.1]. Mukai’s sign convention, which we will follow, is different and we would regard the right hand side as minus the Mukai pairing. Proof. The statement is essentially the definition of the Chern character as a class in HH0 (X) (see [C2, Sec. 6.2] and [C3, Theorem 4.5]).  Given a scheme S and an object x ∈ D b (S), denote by x∨ := RHom(x, OS ) ∈ D b (S) its dual object. Let µx : OS → x∨ ⊗ x

be the natural morphism and

tr : x∨ ⊗ x → OS

the trace morphism ([Mu1], page 114). The following identity holds. (3.18)

tr ◦ µx = rank(x) · 1.

Assume next that S is smooth and projective. Consider the trace pairing ◦

tr

Hom(x, x[i]) ⊗ Hom(x[i], x ⊗ ωS [dim S]) → Hom(x, x ⊗ ωs [dim S]) → H dim S (S, ωS ) ∼ = C, where the left arrow is composition. Mukai shows that the above pairing is a perfect pairing, for 0 ≤ i ≤ dim S ([Mu1], page 114). Mukai’s trace pairing is induced by Serre’s duality as follows. Set y := x∨ ⊗ x. We can rewrite Mukai’s pairing as (3.19)

H i (y) ⊗ H dim S−i (y ⊗ ωS ) → H dim S (S, ωS ),

while Serre’s duality yields a pairing (3.20)

H i (y ∨ ) ⊗ H dim S−i (y ⊗ ωS ) → H dim S (S, ωS ).

Mukai interprets the composition ◦ tr RH om(RH om(x, x)∨ , RH om(x, x)) ∼ = RH om(x, x)⊗RH om(x, x) → RH om(x, x) → OS

INTEGRAL TRANSFORMS AND DEFORMATIONS

35

as an isomorphism ψ : RH om(x, x)∨ → RH om(x, x), or equivalently, ψ : y ∨ → y. Relating the leftmost factors in (3.19) and (3.20) via ψ relates Mukai’s trace pairing to Serre’s duality. Let G : D b (S) → D b (S) be the functor of tensorization by the object x. The right and left adjoints of G are both isomorphic to the functor G† : D b (S) → D b (S), of tensorization with x∨ . Let ∆S : S → S × S be the diagonal morphism. Then ∆S∗ (µx ) : ∆S∗ (OS ) → ∆S∗ (x∨ ⊗ x) induces the unit natural transformation µx : id → G† G for the adjunction. The morphism ∆S∗ (tr) : ∆S∗ (x∨ ⊗ x) → ∆S∗ (OS ) induces the counit natural transformation tr : G† G → id. The morphisms (3.21) µx : Hom(OS , OS [i]) → Hom(OS , x∨ ⊗ x[i]) ∼ = Hom(x, x[i]), G : Hom(OS , OS [i]) → Hom(G(OS ), G(OS )[i])

are equal under the identification x = G(OS ), by [Hu1, Lemma 1.21]. Remark 3.15. If Hom(x, x) is one-dimensional, then (3.22)

tr

Hom(x, x ⊗ ωS [dim S]) → H dim S (S, ωS )

is an isomorphism. Indeed, both spaces are one dimensional and the statement reduces to the non-vanishing of tr : Hom(x, x)⊗Hom(x, x⊗ωS [dim S]) → H dim S (S, ωS ), which holds it being a perfect pairing. Now let t be an element of H dim S (S, ωS ). As a consequence of the above isomorphism, we see that the element µx (t) of Hom(x, x ⊗ ωS [dim S]) vanishes, if rank(x) = 0 and Hom(x, x) is one-dimensional. Indeed, tr(µx (t)) = rank(x) · t = 0 in this case. Let X be a projective K3 surface, M := MH (v) a moduli space of H-stable sheaves with Mukai vector v satisfying the hypothesis of Theorem 2.2 and ΦU : D b (X) → D b (M ) the faithful functor in Theorem 2.2. Assume that (v, v) ≥ 2. Given objects x and y in D b (X), set Hom• (x, y) := ⊕i Hom(x, y[i])[−i], as an object of the derived category of a point. Let Y (x) := Hom• (x, x) be the Yoneda algebra. The morphism µx induces the natural algebra homomorphism µx : Y (OX ) −→ Y (x),

for every object x of D b (X). Define

µΦU (x) : Y (OM ) −→ Y (ΦU (x))

similarly. Let tX ∈ H 2 (X, OX ) be a non-zero element. Let ϕU : H ∗ (X, C)

H ∗ (M, C) √ √ be the homomorphism induced by the correspondence tdX ch(U ) tdM ∈ H ∗ (X × M, Q). Given a ∈ H ∗ (X, C), denote by [ϕU (a)]2 the graded summand in H 2 (M, C). −→

Proposition 3.16. For every object x of D b (X), the following equality holds:

(3.23)

tr (ΦU (µx (tX ))) = rank(x)[ϕU (tX )]2 ,

where tX on the right hand side is considered as an element of the summand H 0,2 (X) of H 2 (X, C), via the Hodge decomposition, and the left hand side is similarly considered as an element of the subspace H 0,2 (M ) of H 2 (M, C). ∗ (x). So Φ(a) = Proof. Let Φ : D b (X) → D b (M ) be the integral functor with kernel U ⊗ πX ΦU (x ⊗ a). Then ΦU (µx (tX )) = Φ(tX ), by the equality of the two homomorphisms displayed in Equation (3.21). Let ϕ : H ∗ (X, C) −→ H ∗ (M, C)

36

E. MARKMAN AND S. MEHROTRA

√ √ ∗ x)π ∗ ∗ tdX ch(U ⊗ πX be the homomorphism induced by the correspondence πX M tdM . Note that ch(x)tX = v(x)tX = rank(x)tX in H ∗ (X, C). So we get the equality ϕ(tX ) = ϕU (ch(x)tX ) = rank(x)ϕU (tX ). It remains to prove the equality (3.24)

tr (Φ(tX )) = [ϕ(tX )]2 .

Let η and ǫ be the unit and counit for the adjunction ∆∗M ⊣ ∆M∗ . We get the following morphisms OM = ∆∗M OM ×M

  ∆∗M ηOM ×M

−→

∆∗M ∆M∗ ∆∗M OM ×M

ǫ[∆∗

M

OM ×M ]

−→

∆∗M OM ×M = OM ,

which compose to the identity. Let ηM : H 2 (M, OM ) → HH−2 (M ) be the composition H (M, OM ) ∼ = Hom(OM , OM [2]) 2

  ∆∗M ηOM ×M

−→ =

Hom(OM [−2], ∆∗M ∆M∗ ∆∗M OM ×M ) Hom(OM [−2], ∆∗M ∆M∗ OM ) = HH−2 (M ).

The analogous homomorphism ηX : H 2 (X, OX ) → HH−2 (X), for the K3 surfaces X, is an isomorphism. Let ǫM : HH−2 (M ) → H 2 (M, OM ) be the morphism induced by ǫ[∆∗M OM ×M ] . We have the following diagram, where the middle square commutes by Theorem 2.15. ǫM

HH−2 (X)

Φ∗

♥7 ηX ♥♥♥♥ ♥ ♥♥ ♥♥♥

tX ∈ H 0,2 (X)

PPP PPP PP = PPPP '

/ HH−2 (M )

I˜∗M

I˜∗X



HΩ−2 (X)

ϕ

 / HΩ−2 (M )

j❚❚❚❚ ❚❚❚❚ηM ❚❚❚❚ ❚❚❚ 

H 0,2 (M ) ∋ tr(Φ(tX )).

❥4 ❥❥❥❥ ❥ ❥ ❥ ❥❥❥❥ 0,2 ❥❥❥❥ π

Here Φ∗ is the homomorphism on Hochschild homology recalled in Equation (2.14). The triangle on the right (with arrow ǫM ) commutes as well. Indeed, the composition Ie

pr 0

M ∆∗M ∆M,∗ ∆∗M OM ×M −→ ⊕Ωi [i] −→ ∆∗M OM ,

where pr 0 is the projection onto the component in degree 0 and I˜M is given in equation (2.13), is nothing but ǫ[∆∗M OM ×M ] . Furthermore, ǫM ◦ ηM is the identity, since ǫ[∆∗M OM ×M ] ◦ ∆∗M ηOM ×M = idOM . Equality (3.24) reduces to the equality h i (3.25) tr (Φ(tX )) = π 0,2 Ie∗M (Φ∗ (ηX (tX ))) . Equivalently, it suffices to prove that ǫM maps ηM [tr (Φ(tX ))] and Φ∗ (ηX (tX )) to the same element of H 2,0 (M ). Let (Φ∗ )† be the adjoint of Φ∗ with respect to the Serre Duality pairing, given in equation (2.16). We will verify equality (3.25) by establishing the equality (3.26)

hν, ηM [tr (Φ(tX ))]i = h(Φ∗ )† ν, ηX (tX )i,

for every element ν ∈ HH−2 (M )∨ , which is in the image of the following composition

∆ M∗ H 2n−2 (M, ωM ) ∼ = HH−2 (M )∨ , = Hom(OM , ωM [2n − 2]) −→ Hom(O∆M , ω∆M [2n − 2]) ∼

INTEGRAL TRANSFORMS AND DEFORMATIONS

37

where the right isomorphism is Serre’s duality. This suffices because h∆M∗ (˜ ν ), λi = h˜ ν , ǫM (λ)i, for every λ ∈ HH−2 (M ) and ν˜ ∈ H 2n−2 (M, ωM ). The following three observations explain the latter equality. (i) The right isomorphism in the displayed composition above is given also by the Mukai pairing under the identification HH2 (M ) ∼ = Hom(O∆M , ω∆M [2n − 2]), by [C2, Subsection 4.11]. (ii) The modified HKR isomorphism Ie∗M is an isometry with respect to the Mukai pairings on HH∗ (M ) and HΩ∗ (M ) (see the conjecture in [C3, Sec. 1.8] and its proof in [HN, Theorem 0.5]). (iii) The compositon of the map Ie∗M ◦∆M,∗ : H i (M, ωM ) → HΩ2n−i (M ) with the projection HΩ2n−i (M ) → H i (M, ωM ) is the identity (see the proof of [HN, Prop. 2.1]). Let ν˜ be an element of H 2n−2 (M, ωM ) mapping to ν. The morphisms Φ(tX ) : Φ(OX ) → Φ(OX )[2] and µΦ(OX )[2] (˜ ν ) : Φ(OX )[2] → Φ(OX ) ⊗ ωM [2n] compose to yield the morphism µΦ(OX )[2] (˜ ν ) ◦ Φ(tX ) : Φ(OX ) → Φ(OX ) ⊗ ωM [2n]. For any two objects x1 , x2 of D b (X), we have the homomorphisms Φ : Hom(x1 , x2 )



Hom(Φ(x1 ), Φ(x2 ))

and its left adjoint with respect to the Serre Duality pairing Φ†L : Hom(Φ(x2 ), Φ(x1 ) ⊗ ωM [2n])



Hom(x2 , x1 ⊗ ωX [2]).

The morphism µΦ(OX )[2] (˜ ν ) belongs to Hom(Φ(x2 ), Φ(x1 ) ⊗ ωM [2n]), for x1 = OX and x2 =  †  OX [2]. Hence, ΦL µΦ(OX )[2] (˜ ν ) belongs to Hom(OX , OX ⊗ ωX ). The equality       ν ) ◦ tX ν ) ◦ Φ(tX ) = T rX Φ†L µΦ(OX )[2] (˜ (3.27) T rM tr µΦ(OX )[2] (˜ is established in [C2, Prop. 3.1]. We prove next that the left hand sides of equations (3.26) and (3.27) are equal. The equality   tr µΦ(OX )[2] (˜ ν ) ◦ Φ(tX ) = ν˜ ◦ tr (Φ(tX ))

holds, by [C2, Lemma 2.4]. Now

T rM (˜ ν ◦ tr (Φ(tX )))

=

T rM ×M (ν ◦ ηM [tr (Φ(tX ))]) ,

by Lemma 3.13, applied with X = M , Y = M × M , f = ∆M , x = OM [−2], y = OM ×M , t = tr (Φ(tX )), and φ = ν˜. Hence, the left hand sides of equations (3.26) and (3.27) are equal. It remains to prove that the right hand sides of equations (3.26) and (3.27) are equal. The following relation between Φ†L and (Φ∗ )† holds, for every object F of D b (X) h i  Φ†L µΦ(F ) (˜ ν) = (Φ∗ )† (∆M∗ (˜ ν )) , F

by [C2, Prop. 3.1]. Taking F = OX , we get h i  (3.28) Φ†L µΦ(OX ) (˜ ν) = (Φ∗ )† (∆M∗ (˜ ν ))

OX

=

h

i (Φ∗ )† (ν)

OX

.

In addition, we have the following relation between the Serre Duality pairing for morphisms in D b (X) and the Serre Duality pairing for morphisms in D b (X × X), or for natural transformation. h i (3.29) h(Φ∗ )† (ν), ηX (tX )i = h (Φ∗ )† (ν) , tX i. OX

38

E. MARKMAN AND S. MEHROTRA

Indeed, the morphism (Φ∗ )† (ν) belongs to Hom(∆X∗ OX , ∆X∗ ωX ). Hence, (Φ∗ )† (ν) = ∆X∗ (φ), for a morphism φ in Hom(OX , ωX ). Now apply Lemma 3.13 with Y = X × X, f = ∆X , x = OX [−2], y = OX×X , t = tX , and φ as above, to obtain Equation (3.29). Combining the last two equations above we get: h i  (3.29) (3.28 ) h(Φ∗ )† (ν), ηX (tX )i = h (Φ∗ )† (ν) ν ) , tX i. , tX i = hΦ†L µΦ(OX ) (˜ OX

This is precisely the desired equality of the two right hand sides of equations (3.26) and (3.27). 

Example 3.17. Let us verify Equation (3.23) in a simple case. Take n = 1, identify X with M := X [1] , and set U := I∆ to be the ideal sheaf of the diagonal in X × X [1] . Choose a sheaf F on X satisfying hi (F ) = 0, for i > 0. Consider the short exact sequence 0 → U → OX×X → O∆X → 0.

Then rank(ΦU (F )) = χ(F ) − rank(F ) and ΦU (F ) fits in the exact triangle F [−1] → ΦU (F ) → H 0 (F ) ⊗C OX → F.

Furthermore, [ϕU (tX )]2 = −tX , under the identification of X with M := X [1] . The FourierMukai transform ΦOX×X with respect to the structure sheaf OX×X sends µF (tX ) : F → F [2] to zero. Indeed, consider the cartesian diagram π2

X ×X

/X

π1

κ



X

κ

 / {pt}.

Then Rπ2∗ π1∗ (µF (tX )) = κ∗ Rκ∗ (µF (tX )) = κ∗ κ∗ (µF (tX )) and the morphism κ∗ (µF (tX )) : H 0 (F ) → H 0 (F )[2] in D b ({pt}) vanishes. We get the commutative diagram / H 0 (F ) ⊗C OX

ΦU (F )

ΦOX×X (µF (tX ))=0

ΦU (µF (tX ))



ΦU (F )[2]

/F µF (tX )



/ H 0 (F ) ⊗C OX [2]

 / F [2].

We see that indeed tr(ΦU (µF (tX ))) = −rank(F )tX = rank(F )[ϕU (tX )]2 . Note, by the way, that ΦU : Hom(F, F [2]) → Hom(ΦU (F ), ΦU (F )[2]) is an isomorphism. If the sheaf F is simple, then Hom(ΦU (F ), ΦU (F )[2]) is one-dimensional. In that case we get the equality, rank(ΦU (F ))ΦU (µF (tX )) = rank(F )µΦU (F ) ([ϕU (tX )]2 ), since both sides above have the same trace. 3.6. A relation in the Yoneda algebra of ΦU (x). Let tX be a non-zero element of H 2 (X, OX ) and ϕU : H ∗ (X, C) → H ∗ (M, C) the homomorphism in Proposition 3.16. Set (3.30)

tM := [ϕU (tX )]2 .

Lemma 3.18. The class tM spans H 2,0 (M ). Furthermore, the following equality holds, for every object x ∈ D b (X). (3.31)

tr (ΦU (µx (tX )))

= rank(x)tM .

Proof. The statement follows immediately from Theorem 2.16 and Proposition 3.16.



INTEGRAL TRANSFORMS AND DEFORMATIONS

39

We regard tX also as an element of the subspace Ext2 (OX , OX ) of Y (OX ). Given an object x of D b (X), set tx := µx (tX ) tΦU (x) := µΦU (x) (tM ).

√ Let v(x) := ch(x) tdX be the Mukai vector of x. Then tΦU (x) is an element of Ext2 (ΦU (x), ΦU (x)) satisfying      (3.32) tr tΦU (x) = tr µΦU (x) (tM ) = rank(ΦU (x)) · tM = − v, v(x)∨ tM .

Lemma 3.19. Let x be an object of D b (X) satisfying Hom(x, x) ∼ = C. The following equation 2n holds in Ext (ΦU (x), ΦU (x)). n−1 n (3.33) − (v, v(x)∨ ) tΦU (x) ΦU (tx ) = rank(x) tΦU (x) .

Proof. The vector space Ext2n (ΦU (x), ΦU (x)) is dual to Hom(ΦU (x), ΦU (x)), which is isomorphic to Hom(x, x), by Theorem 2.2, and is thus one-dimensional. Consequently, the two sides of Equation (3.33) are linearly dependent. Equation (3.33) would thus follow from Equations (3.31) and (3.32), once we prove that the Yoneda product n−1 tΦU (x) ◦ : Ext2 (ΦU (x), ΦU (x)) → Ext2n (ΦU (x), ΦU (x)) factors through tr : Ext2 (ΦU (x), ΦU (x)) → H 2 (M, OM ). We prove this factorization next. Note that the morphism µΦU (x) : OM → ΦU (x)∨ ⊗ ΦU (x) is compatible with the Yoneda n−1  n−1 product. Hence, tΦU (x) = µΦU (x) tM . For every object y of D b (M ), and for every integers i and j, the outer square of the following diagram is commutative. / Exti+j (y, y) 5 ❦ ❦❦ α ❦❦❦❦ ❦ ❦❦❦ ❦❦❦

Exti (y, y) ⊗ Extj (y, y) O

id⊗µ

Exti (y, y) ⊗ H j (OM )

tr

tr⊗id



H i (OM ) ⊗ H j (OM )

 / H i+j (OM ).

The homomorphism α, defined to make the diagram commutative, factors through the bottom left vertical homomorphism tr ⊗ id, whenever the right vertical trace homomorphism is an n−1 of H 2n−2 (OM ), isomorphism. Apply it with y = ΦU (x), i = 2, j = 2n − 2, the element tM and observe that the trace homomorphism tr : Ext2n (ΦU (x), ΦU (x)) → H 2n (OM ) is an isomorphism, by Remark 3.15, and the fact that Hom(ΦU (x), ΦU (x)) ∼ = Hom(x, x) ∼ = C. The Equality (3.33) now follows, where the coefficient on its left hand side is explained by the equality rank (ΦU (x)) = − (v, v(x)∨ ).  Both sides of Equation (3.33) vanish, whenever rank(x) = 0 or rank(ΦU (x)) = 0 (that is − (v, v(x)∨ ) = 0). This can be seen directly, without using the above lemma, as follows. The left hand side vanishes if rank(x) = 0, since tx := µx (tX) vanishes, by Remark 3.15. The right n hand side vanishes if rank(ΦU (x)) = 0, since tΦU (x) := (µΦU (x) (tM ))n = µΦU (x) ((tM )n ) vanishes, by Remark 3.15 again. Assume next that Hom(x, x) is one-dimensional. Let trx−1 be the inverse of the isomorphism given in Equation (3.22). If rank(x) and rank(ΦU (x)) do not vanish, Equation (3.33) is equivalent to the following equation: n−1 −1 (3.34) tΦU (x) (tnM ) . ΦU (trx−1 (tX )) = trΦ U (x)

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We will verify Equation (3.34) assuming only that rank(ΦU (x)) does not vanish (Theorem 3.21). We expect the above equation to hold even if rank(ΦU (x)) vanishes. 3.7. The natural transformation h2 is the Mukai vector. Set R(M ) := Ext2n (OM , OM )[−2n], regarded as an object of D b (pt). The objects of the exact triangle displayed in Equation (3.6) correspond to kernels of integral endo-functors of D b (X). The object R corresponds to the functor of tensorization by R(M ) over C. The object Y := O∆X ⊗C Y (OM ) corresponds to the functor of tensorization by Y (OM ) over C. The object A is the kernel of the functor ΨU ΦU . The morphisms of the exact triangle (3.6) correspond to natural transformations between these endo-functors. Given objects F1 , F2 of D b (X), we get the short exact sequence (3.35) qF

hF

2 2 0 → Hom (F1 , F2 ⊗C R(M )) −→ Hom (F1 , F2 ⊗C Y (OM )) −→ Hom (F1 , ΨU ΦU (F2 )) → 0.

Exactness of the above sequence follows from the splitting of the exact triangle (3.6). Equivalently, we have the short exact sequence hF

0 → Hom (F1 , F2 [−2n]) ⊗ Ext2n (OM , OM ) →2 ⊕ni=0 Hom (F1 , F2 [−2i]) ⊗ Ext2i (OM , OM ) Q

→ Hom (ΦU (F1 ), ΦU (F2 )) → 0.

Set Y 2k := H 2k (M, OM ), so that Y 2k [−2k] is the graded summand of Y (OM ) of degree 2k. Let tX ∈ H 2 (X, OX ) be a non-zero class, tM ∈ H 2 (M, OM ) the class associated to tX in ˜0 ⊗ 1 + h ˜ 2 ⊗ t∗ + Equation (3.30), and let tF be the class µF (tX ) in Ext2 (F, F ). Write h = h M ˜ 4 ⊗ (t∗ )2 , using the notation of Theorem 3.10. Above, h ˜ 2j is a natural transformation from h M the identity functor 1 of D b (X) to 1[2j]. Theorem 3.20. Let F1 and F2 be objects of D b (X). (1) The following is a short exact sequence Q

h

0 → Hom• (F1 , F2 [−2n]) ⊗ Y 2n → Hom• (F1 , F2 ) ⊗ Y (OM ) → Hom• (ΦU (F1 ), ΦU (F2 )) → 0,

where h and Q are homomorphisms of degree 0, and Q(g ⊗ y) = µΦU (F2 ) (y)ΦU (g). (2) If F1 and F2 are sheaves on X, then Hom(ΦU (F1 ), ΦU (F2 )[k]) = 0, for k > 2n and for k < 0. The homomorphism Q restricts to an isomorphism for degrees in the range 0 ≤ k ≤ 2n − 1. In degree 2n we get the short exact sequence   Ext2 (F1 , F2 ) ⊗ Y 2n−2 Q h → 0 → Hom(F1 , F2 ) ⊗ Y 2n →  Hom (ΦU (F1 ), ΦU (F2 )[2n]) → 0, ⊕ Hom(F1 , F2 ) ⊗ Y 2n where h is given by the equality ˜2 h(f ⊗ tn ) = (h M

F2

n−1 ˜ 0 ◦ f ) ⊗ tn , + (h ◦ f ) ⊗ tM M F2

for all f ∈ Hom(F1 , F2 ). Consequently, if in addition Hom(F1 , F2 ) = 0, then Q induces an isomorphism in degree 2n as well. (3) When F is a simple sheaf on X, the kernel of   Ext2 (F, F ) ⊗ Y 2n−2  −→ Hom (ΦU (F ), ΦU (F )[2n]) ⊕ Q :  2n Hom(F, F ) ⊗ Y is spanned by the element

(3.36)

n−1 − 1 ⊗ tnM , − (v, v(F ∨ ))trF−1 (tX ) ⊗ tM

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41

where trF : Ext2 (F, F ) → H 2 (X, OX ) is the isomorphism given in Equation (3.22). Proof. (1) The short exact sequence in formula (3.35) establishes the statement in degree zero. For degree k, replace F2 by the object F2 [k]. We have the equalities   ∗ ∗ ∗ ∗ F (π ∗ F ) (y)◦ΞU (π (g)) = µΦ (F ) (y)◦ΦU (g), (y)) ◦ π (g) = µΞU (πX (qF2 )∗ (g⊗y) = ΞU µπX M X X 2 2 2 U

where the first equality follows by Lemma 3.11, the second is due to the fact that the kernel ∆23∗ (U ) in D b ((X × M ) × M ) of the integral functor ΞU is supported on the diagonal, and ∗ . The formula for Q(g ⊗ y) follows. the third since ΦU = ΞU ◦ πX j (2) Note first that Ext (F1 , F2 ) vanishes, for j 6∈ {0, 1, 2}, since F1 and F2 are sheaves. When F1 and F2 are sheaves, and k > 2n or k < 0, then the degree k component of h is an injective homomorphism from a finite dimensional vector space to itself, by part 1, hence an isomorphism. If k is in the range 0 ≤ k ≤ 2n − 1, the space Hom(F1 , F2 [k − 2n]) ⊗ Y 2n vanishes, hence the degree k component of Q is an isomorphism. The statement in degree 2n follows easily from part 1. (3) If rank(F ) 6= 0, then the element given in (3.36) belongs to the kernel of m2n , by Equations (3.18) and (3.33). This element must span the kernel, as the kernel is one-dimensional, by part 2. The kernel is spanned by ˜ 0 ⊗ tn + h ˜ 2 ⊗ tn−1 h(1F ⊗ tn ) = h M

F

M

F

M

˜ 0 (v, v(F ∨ ))tr −1 (tX ), h F F

˜2 = for every simple sheaf F satisfying the inas well. Hence, h F ˜ 0 does not vanish and we normalize the natural equality rank(F ) 6= 0. We conclude that h ˜ 0 = −1. Then transformation h by rescaling it, so that h F ˜ 2 = −(v, v(F ∨ ))tr −1 (tX ), (3.37) h F

F

for every simple object F satisfying the inequality rank(F ) 6= 0. Let σX ∈ H 0 (X, ωX ) be the ˜ 2 ⊗ σX belongs to HH0 (X) and the class, such that T rX (tX ⊗ σX ) = 1. The class h2 := h ∨ above equation translates to T rX (h2F ) = −(v, v(F )). The latter equality holds for all simple sheaves F satisfying rank(F ) 6= 0. This suffices to determines the algebraic part of the class √ Ie∗X (h2 ) in HΩ0 (X) as ch(v) tdX , by Lemma 3.14. Hence, Equality (3.37) holds regardless of the vanishing of rank(F ). Part (3) of the Theorem follows. 

Proof of part 2 of Theorem 3.10. Write H 1,1 (X, C) = [H 1,1 (X, Z) ⊗Z C] ⊕ Θ(X), where Θ(X) is the transcendental subspace. Both ch(v) and I∗X (h2 ) are elements of H 0 (X, C) ⊕ H 1,1 (X, C) ⊕ H 4 (X, C).

Now ch1 (v) belongs to H 1,1 (X, Z) ⊗Z C and Equation (3.37) in the proof of Theorem 3.20 (3) implies that the projection of I∗X (h2 ) to H 1,1 (X, Z) ⊗Z C is equal to ch1 (v). Varying the surface X in a codimension one family in moduli keeping ch1 (v) of Hodge type (1, 1), the classes ch(v) and I∗X (h2 ) define two continuous sections of the Hodge bundle H 1,1 with fiber H 1,1 (X, C), the difference of which is purely transcendental. But a purely transcendental continuous section of H 1,1 over such a family must vanish, by the density of Hodge structures with trivial transcendental subspace.  Let F be a simple sheaf on X and Y (F ) := ⊕2i=0 Exti (F, F )[−i] its Yoneda algebra. Let Y (ΦU (F )) := ⊕i∈Z Hom(ΦU (F ), ΦU (F )[i])[−i] be the Yoneda algebra of ΦU (F ). Let f : Y (F ) ⊗C C[z] → Y (ΦU (F ))

be the algebra homomorphism restricting to Y (F ) ⊗ 1 as ΦU and sending the indeterminate z to µΦU (F ) (tM ) in Hom(ΦU (F ), ΦU (F )[2]). The homomorphism f is well defined, since

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µΦU (F ) (tM ) belongs to the center of Y (ΦU (F )), as tM is a natural transformation from 1Db (M ) to 1Db (M ) [2]. Theorem 3.21. The homomorphism f is surjective and its kernel is the ideal generated by (3.38)

1 ⊗ z n + (v, v(F ∨ ))trF−1 (tX ) ⊗ z n−1 .

 Note that the equality f (1 ⊗ z n ) = −f (v, v(F ∨ ))trF−1 (tX ) ⊗ z n−1 implies the vanishing of f (1 ⊗ z n+1 ), since (trF−1 (tX ))2 = 0. Hence, f factors through Y (F ) ⊗ Y (OM ).

Proof. The algebra Y (F )⊗C C[z] is graded, where z has degree 2. Let Y ′ be its quotient by the ideal generated by the homogeneous element (3.38). The homomorphism f factors through a homomorphism f¯ : Y ′ → Y (ΦU (F )), by Theorem 3.20 (3). The algebra Y ′ is graded. Denote by Yd′ its graded summand of degree d. Then  dim Ext1 (F, F ) if d is odd and 0 < d < 2n,    2 if d is even and 0 < d < 2n, dim(Yd′ ) = 1 if d = 0 or d = 2n,    0 otherwise.

These are precisely the dimensions of the graded summands of Y (ΦU (F )), by Theorem 3.20 (2). Hence, it suffices to prove that the homomorphism f¯ is surjective. Surjectivity follows from Theorem 3.20 (2).  3.8. Moduli spaces of sheaves over the moduli space M . We continue to assume that M := MH (v) and the modular object A over X × X is totally split as in Definition 3.1. Let S be a scheme of finite type over C. A coherent OS -module F is simple, if EndS (F, F ) is spanned by the identity. Let Spl(S) be the moduli space of simple sheaves over S [AK]. Denote by [F ] the point of Spl(S) corresponding to F . Corollary 3.22. Let F be a simple coherent OX -module and assume that ΦU (F )[i] is equivalent to a coherent OM -module FM , for some integer i. Then FM is simple. The functor ΦU induces an isomorphism, from the open subset U := {[F ] : ΦU (F )[i] is equivalent to a coherent OM -module}

of Spl(X), onto an open subset of Spl(M ).

Proof. If F is simple, then it is a smooth point of the moduli space Spl(X) of simple sheaves [Mu1]. The functor ΦU induces isomorphisms ΦU : ExtiX (F, F ) → ExtiM (FM , FM ), for i = 0, 1, by Theorem 3.20. Hence, FM is simple, if it is a sheaf. The functor ΦU defines a morphism φ : U → Spl(M ). This follows from a flatness result for Fourier-Mukai functors ([Mu2], Theorem 1.6). The differential of φ at [F ] is the isomorphism of Zariski tangent spaces ΦU : Ext1X (F, F ) → Ext1M (FM , FM ). Combining the injectivity of the differential with the smoothness of Spl(X), we conclude that the dimension D of Spl(M ) at [FM ] is larger than or equal to the dimension d of Spl(X) at [F ]. The surjectivity of the differential implies that D ≤ d. Thus, D = d = dim Ext1 (FM , FM ) , and Spl(M ) is smooth at [FM ]. The homomorphism ΦU : Hom(F1 , F2 ) → Hom(Φ(F1 ), Φ(F2 )) is an isomorphism, for any two sheaves F1 , F2 on X, by Theorem 3.20. It follows that φ is injective and hence an isomorphism onto its image.  Example 3.23. Choosing F to be a sky-scraper sheaf of a point of the K3 surface X we see that X is a connected component of Spl(M ). The twisted universal sheaf U over X × M for the coarse moduli space M := MH (v) of H-stable sheaves over the K3 surface X is also a universal sheaf of X as a moduli space of sheaves over M .

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43

Example 3.24. Let ΦU ∨ : D b (X) → D b (M ) be the integral functor defined by replacing the kernel U with U ∨ in equation (1.1). Define ΨU ∨ : D b (M ) → D b (X) similarly. Note that the n−1 ∆∗ OX [−2i], by Theorem kernel of the integral functor ΨU ∨ ◦ ΦU ∨ is also isomorphic to ⊕i=0 2.2, as its kernel is the pullback of the kernel A of ΨU ◦ ΦU via the transposition τ of the two factors of X × X. Now A is τ -invariant. Theorem 3.20 applies to the functor ΦU ∨ as well. Let w ∈ K(X) be another class and H ′ a w-generic polarization, such that MH ′ (w) is smooth and projective. Assume that all sheaves parametrized by one of MH (v) or MH ′ (w) are locally free.6 Denote by wn the class of w ⊗ [H ′⊗n ]. Then MH ′ (w) is isomorphic to MH ′ (wn ). Assume that both v and w have positive rank. Fix n sufficiently negative. Then Exti (F, F ′ ) is isomorphic to H i (F ∗ ⊗ F ′ ) if F is locally free and to H 2−i (F ⊗ (F ′ )∗ )∗ if F ′ is locally free, and it thus vanishes, for i 6= 2, for all H-stable sheaves F of class v, and for all H ′ -stable sheaves F ′ of class w. Thus, ΦU ∨ (F ′ )[2] is equivalent to a coherent sheaf, for all such F ′ . We get a component of Spl(MH (v)) isomorphic to MH ′ (w), via Corollary 3.22. 4. The transposition of the factors of M × M Let X be a K3 surface. If X is projective we assume given a primitive Mukai vector v with c1 (v) of Hodge-type (1, 1) and a v-generic polarization H and we set M := MH (v). We also consider the case where X is a K¨ ahler non-projective K3 surface and M = X [n] . Let F be the modular object and E the modular sheaf over M × M (Definition 1.5). We calculate the extension sheaves E xti (E , OM ×M ) and E xti (E , E ) and the torsion sheaves T ori (E ∗ , E ) in this section. Let τ : M × M → M × M be the transposition of the two factors.

Lemma 4.1. The following two objects of D b (M × M ) are isomorphic. (4.1) F ∼ = τ ∗ F ∨ [2].

∗ U ∨ ⊗ π ∗ U [2]). We have the isomorphisms: Proof. The object F is defined as Rπ13∗ (π12 23   ∗ ∨ ∗ ∨ ∼ U [2] , OM ×M F = RH om Rπ13∗ π12 U ⊗ π23 o n  ∗ ∗ ! ∼ U ∨ ⊗ π23 U [2], π13 OM ×M = Rπ13∗ RH om π12  ∗ ∗ ∼ U ⊗ π23 U∨ ∼ = Rπ13∗ π12 = τ ∗ F [−2],

where the fist isomorphism is clear, the second is Grothendieck-Verdier Duality, the third uses the triviality of ωπ13 , and the last is clear.  Let ∆ ⊂ M × M be the diagonal. Pushforward via the inclusion morphism embeds the category of coherent sheaves on ∆ in that of M ×M . We suppress the pushforward notation and regard the former as a subcategory of the latter. Recall that E := H −1 (F ), H 0 (F ) ∼ = O∆ , and all other sheaf cohomologies of F vanish. The following is a more precise description of E due to the first author. Proposition 4.2 ([Ma5], Proposition 4.5). The reflexive sheaf E is locally free away from ∆. Its restriction to ∆ is untwisted, and is described as follows: (i) E ⊗ O∆ ∼ = Ω2∆ /O∆ · σ, where σ is the symplectic form; (ii) T ori (E , O∆ ) ∼ = Ωi+2 ∆ , for i > 0. Set E ∨ := RH om(E , OM ×M ) and E ∗ := H om(E , OM ×M ), so that E ∗ := H 0 (E ∨ ). 6 If v = (r, c, s) is primitive, H is v-generic, r ≥ 2, (r, c, s + 1) = ku, where k is an integer and u is a primitive

Mukai vector satisfying (u, u) ≤ −4, then all sheaves parametrized by MH (v) are locally free.

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Lemma 4.3. The following sheaves isomorphisms exist: E∗ ∼ = τ ∗E , E xt1 (E , OM ×M ) ∼ = O∆ , E xt2n−2 (E , OM ×M ) ∼ = O∆ ,

and E xti (E , OM ×M ) ∼ = 0, if i does not belong to {0, 1, 2n − 2}. Proof. Dualizing the exact triangle E [1] → F → O∆ → E [2], we get the exact triangle E ∨ [−2] → ω∆ [−2n] → F ∨ → E ∨ [−1].

Trivializing ω∆ , shifting by 2, and applying the isomorphism (4.1), we get the following exact triangle. α E ∨ → O∆ [2 − 2n] → τ ∗ F → E ∨ [1]. The sheaf homomorphisms H i (α) : H i (O∆ [2 − 2n]) → H i (τ ∗ F ) vanish for all i, since n ≥ 2. The long exact sequence of sheaf cohomology breaks into H −1 (τ ∗ F ) ∼ = H 0 (E ∨ ), 0 ∗ 1 ∨ 2n−2 ∨ i ∨ ∼ ∼ H (τ F ) = H (E ), and H (E ) = O∆ , and H (E ) = 0, if i does not belong to {0, 1, 2n − 2}.  The following proposition is used in the proof of [MM2, Theorem 1.11]. Proposition 4.4. The sheaf E xti (E , E ) comes with a decreasing filtration F 0 E xti (E , E ) ⊂ F −1 E xti (E , E ) ⊂ · · · ⊂ F 2−2n E xti (E , E ) = E xti (E , E )

p,q with graded pieces E∞ := F q E xtp+q (E , E )/F q+1 E xtp+q (E , E ) satisfying 0,0 (1) E∞ = F 0 E xt0 (E , E ) ∼ = E ∗ ⊗ E /torsion, 1,−1 ∼ 3 E∞ = Ω∆ , 2n−2,2−2n ∼ 2n E∞ = Ω∆ , and all other graded pieces of the filtration of E xt0 (E , E ) vanish. (2) We have the short exact sequence 2n−1 → 0, 0 → Ω2∆ /(σ) → E xt1 (E , E ) → Ω∆

1,0 2n−2,3−2n where the subsheaf is E∞ and the quotient is E∞ and all other graded pieces 1 of the filtration of E xt (E , E ) vanish. 2n−i , for 2 ≤ i ≤ 2n − 3. (3) E xti (E , E ) ∼ = Ω∆ 2n−2 ∼ E xt (E , E ) = Ω2∆ /(σ), and E xti (E , E ) = 0, for i > 2n − 2. (4) The torsion subsheaf of E ∗ ⊗ E is isomorphic to Ω4∆ . j+4 T orj (E ∗ , E ) ∼ = Ω∆ , for j ≥ 1. In particuler, T orj (E ∗ , E ) vanishes for j > 2n − 4.

Proof. Let Wj → Wj+1 → · · · → W−1 → W0 → E be a locally free resolution of E . Denote the locally free complex by (W• , d) and let (W•∗ , d∗ ) be the dual complex. We get the double complex ∗ W p,q := W−p ⊗ Wq , p ≥ 0, q ≤ 0

and the associated single complex K k := ⊕p+q=k W p,q with differential D, which restricts to W p,q as (−1)p (d∗ ⊗ 1) + (1 ⊗ d) : W p,q → W p+1,q ⊕ W p,q+1 [GH, Ch. 4, Sec. 5]. Note that (W• , d) is quasi-isomorphic to E , (W•∗ , d∗ ) represents E ∨ := RH om(E , OM ×M ), and (K • , D) represents RH om(E , E ) in D b (M × M ). In particular, we have the isomorphism H i (K • , D) ∼ = E xti (E , E ).

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45



Consider the decreasing filtration F q K i := ⊕p+q′ =i,q′ ≥q W p,q and denote the induced filtrap,q tion on sheaf cohomology by F q E xti (E , E ). Set E∞ := F q E xtp+q (E , E )/F q+1 E xtp+q (E , E ). We have a spectral sequence converging to H i (K • , D), with E1p,q := H p ((W•∗ , d∗ ) ⊗ Wq ) ∼ = E xtp (E , Wq ) ∼ = E xtp (E , OM ×M ) ⊗ Wq . E2p,q ∼ = H q (E xtp (E , OM ×M ) ⊗ W• , 1 ⊗ d) ∼ = T or−q (E xtp (E , OM ×M ), E ) .

We have seen that E xtp (E , OM ×M ) is isomorphic O∆ , for p = 1 and p = 2n − 2, and it vanishes for all other positive values of p, by Lemma 4.3. Furthermore, we have the isomorphisms 2−q T or−q (O∆ , E ) ∼ = Ω∆ , for q < 0, and T or0 (O∆ , E ) ∼ = Ω2∆ /(σ), by Proposition 4.2. We get the following table for the E2 term of the spectral sequence. q\p 0 1 ··· 2n-2 0 E∗ ⊗E Ω2∆ /(σ) Ω2∆ /(σ) -1 T or1 (E ∗ , E ) Ω3∆ Ω3∆ -2 T or2 (E ∗ , E ) Ω4∆ Ω4∆ .. . 2-2n

T or2n−2 (E ∗ , E )

Ω2n ∆

Ω2n ∆

2n−2,q We show first that E∞ = E22n−2,q , for all q. For j ≥ 2, the differential of the spectral sequence is dj : Ej2n−2,q → Ej2n−j−1,q+j . The vanishing of the columns other than for p = 2n−2,q 2n−2,q 1,q+2n−2 0, 1, 2n − 2 implies that E22n−2,q = E2n−2 . The differential d2n−2 : E2n−2 → E2n−2 vanishes for all q. Indeed, for q = 2 − 2n the homomorphism is a section of H 0 (Ω2∆ /(σ)), which vanishes. For all other values of q the differential vanishes since its target vanishes. The above vanishing of dj : Ejp,q → Ejp−j+1,q+j , for j ≥ 2 and p ≥ 2, implies also that 1,−1 1,0 E∞ = E21,−1 and E∞ = E21,0 . p,q The sheaves E∞ vanish for p + q < 0, since E xtp+q (E , E ) vanishes for these values. Hence, 1,q 0,q+2 d1,q 2 : E2 → E2

is injective for q ≤ −2 and surjective for q < −2. In particular, the homomorphism d2 : E21,−2 = Ω4∆ → E ∗ ⊗ E

is injective and the sheaves T or−q (E ∗ , E ) are as claimed.



Remark 4.5. Let β : Y → M × M be the blow-up centered along the diagonal. We claim that the sheaf β ∗ E has a non-trivial torsion subsheaf. This is seen as follows. The sheaf E ∗ is isomorphic to β∗ V , for a locally free sheaf V over Y , by [Ma5, Prop. 4.5]. E ∗ ⊗ E has a nontrivial torsion subsheaf, by Proposition 4.4 (4). On the other hand, E ∗ ⊗ E ∼ = β∗ (V ⊗ β ∗ E), ∗ by the projection formula. Hence, V ⊗ β E has a non-trivial torsion subsheaf. We conclude that so does β ∗ E , as claimed. 5. A simple and rigid comonad in D b (M × M ) 5.1. F is simple and rigid. We keep the notation of section 1.1, so (X, H) is a polarized K3 surface, M := MH (v) is a smooth and projective moduli space of H-stable sheaves on X, of dimension 2n, n ≥ 2, U is a universal sheaf over X × M , ΦU : D b (X) → D b (M, θ) is the integral functor with kernel U , and ΨU : D b (M, θ) → D b (X) is its right adjoint. Consider ∗ θ −1 ). the following object in D b (X × M, πM ∗ V := U ∨ ⊗ πX ωX [2].

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Let πij be the projection from M × X × M onto the product of the i-th and j-th factors. The kernel of the endo-functor L := ΦU ΨU : D b (M, θ) → D b (M, θ) is the object ∗ ∗ F = π13∗ (π12 V ⊗ π23 U)

(5.1)

of D b (M × M, π1∗ θ −1 π2∗ θ). We prove in this section the following statement. Lemma 5.1. Assume that M = X [n] and U is the universal ideal sheaf or, more generally, that the morphism α of Theorem 1.1 is an isomorphism. (1) The following isomorphisms hold: Hom(F , F ) ∼ = C, Hom(F , F [k]) = 0, for odd k. (2) More generally, Hom(F , F [k]) is isomorphic to h i⊕j+1 h i⊕i+1 n−2 n−1 HH k+2j+4−4n (X) , HH k−2i (X) ⊕ ⊕j=0 ⊕i=0 for all integers k.

Proof. Part (1): Note that the Hochschild cohomology HH i (X) := HomX×X (O∆X , O∆X [i]) vanishes for i < 0, for i > 4, and for odd i. Part (1) follows from part (2) and the isomorphism HH 0 (X) := HomX×X (O∆X , O∆X ) ∼ = C. Part (2): Let ΦV : D b (X) → D b (M, θ −1 ) be the integral functor with kernel V and let ΨV : D b (M, θ −1 ) → D b (X) be its right adjoint. We get the object ∗ ∗ V ⊠ U := π13 V ⊗ π24 U.

in D b (X × X × M × M, π3∗ θ −1 π4∗ θ). Let

Γ : D b (X × X) −→ D b (M × M, π1∗ θ −1 π2∗ θ)

be the integral functor with kernel V ⊠ U and let Γ†R : D b (M × M, π1∗ θ −1 π2∗ θ) → D b (X × X) be its right adjoint. Then Γ†R is the integral transform with kernel (V ⊠ U )∨ ⊗ π ∗ ωX ⊗ π ∗ ωX [4] ∼ =U ⊠V. 1

2

Note that Γ is the cartesian product of the functors ΦW and ΦU . Similarly, Γ†R is the cartesian product of their right adjoints ΨV and ΨU . Let A be the object of D b (X × X) given in Equation (1.2). We have the following isomorphisms. Γ (O∆ ) ∼ (5.2) = F, X

Γ†R (O∆M )

∼ = A.

The composition ΨU ΦU : D b (X) → D b (X) has kernel A . The kernel of the composition ΨV ΦV : D b (X) → D b (X) is the pull-back of A via the involution of X × X interchanging the two factors. A is invariant under this pull-back, and so A is the kernel of ΨV ΦV as well. We conclude that the composite endo-functor Γ†R Γ : D b (X × X) −→ D b (X × X)

∗ A ⊗ π ∗ A in D b (X × X × X × X). We get the following isomorphism. has kernel π13 24

(5.3)

Γ†R Γ(O∆X )

∼ =

A ◦A.

INTEGRAL TRANSFORMS AND DEFORMATIONS

47

The object A is isomorphic to O∆X ⊗C λn , where λn is an object of D b (pt), by Theorem 2.2 or by the assumption that α is an isomorphism. Hence, the endo-functor Γ†R Γ is isomorphic to the functor of tensorization over C with the object (λn )⊗2 of D b (pt). Now ⊕i+1 ⊕ ⊕n−2 (Opt [2j + 4 − 4n])⊕j+1 . (λn )⊗2 ∼ = ⊕n−1 (Opt [−2i]) i=0

j=0

We get the isomorphisms:

HomM ×M (F , F [k]) ∼ = HomM ×M (Γ(O∆X ), Γ(O∆X )[k])   † ∼ = HomX×X O∆ , Γ Γ(O∆ )[k] X

R

X

 ∼ = HomX×X O∆X , O∆X ⊗C (λn )⊗2 [k] .

Part (2) follows immediately from the isomorphisms above.



5.2. E is simple and rigid. Let E be the sheaf cohomology H −1 (F ), where F is the object of D b (M × M ) given in Equation (5.1). We get the exact triangle (5.4)

β

γ

ǫ

E [1] → F → O∆M → E [2],

where ǫ is the morphism inducing the counit for the adjunction ΦU ⊣ ΨU . The following rigidity result is a crucial ingredient in the proof of Theorem 1.7. Lemma 5.2. Assume that M = X [n] and U is the universal ideal sheaf. Then the sheaf E is simple and rigid. In other words, Hom(E , E ) is one-dimensional and Ext1 (E , E ) vanishes. Proof. Step 1: The integral functor Γ†R : D b (M × M ) → D b (X × X) takes the exact triangle (5.4) to the exact triangle (5.5)

Γ† (β)

m

Γ† (γ)

R R Γ†R (E [1]), A ◦ A −→ A −→ Γ†R (E [1]) −→

where A ◦ A is the convolution and m := Γ†R (ǫ) is the multiplication, by Equations (5.2), (5.3), and the proof of Lemma 5.1. Now m has a right inverse given by η◦1 A ∼ = A ◦ O∆X −→ A ◦ A ,

where η : O∆X → A induces the unit for the adjunction ΦU ⊣ ΨU . Thus, Γ†R (γ) = 0 and the exact triangle (5.5) splits. The left adjoint Γ†L of Γ is isomorphic to a shift of the right adjoint Γ†R , since X and M have trivial canonical line bundles. We conclude that Γ†L (γ) = 0. In particular, the homomorphism Γ†L (β)∗ : Hom(Γ†L (F ), x) −→ Hom(Γ†L (E [1]), x)

is surjective, for all objects x of D b (X × X). Take x = O∆X [k], apply the adjunction Γ†L ⊣ Γ, and use the isomorphism Γ(O∆X ) ∼ = F , to conclude that the homomorphism Hom(F , F [k]) → Hom(E [1], F [k])

is surjective, for all k. We get the inequality (5.6)

dim Hom(E [1], F [k]) ≤ dim Hom(F , F [k]),

for all k. Step 2: Apply the functor Hom(E , •) to the exact triangle (5.4). We get the long exact sequence Hom(E , O∆M [k − 1]) → Hom(E , E [k + 1]) → Hom(E , F [k]) → Hom(E , O∆M [k]).

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When k = −1, the first and fourth terms vanish and so Hom(E , E ) → Hom(E , F [−1]) is an isomorphism. Now dim Hom(E , F [−1]) ≤ dim Hom(F , F ), by the inequality (5.6), and dim Hom(F , F ) = 1, by Lemma 5.1. We conclude that dim Hom(E , E ) = 1. When k = 0, the first term in the long exact sequence above vanishes. The third term vanishes, by the inequality (5.6) and the vanishing of Hom(F , F [1]) established in Lemma 5.1. We conclude that Hom(E , E [1]) = 0.  Assume that M = X [n] as above. Let E : D b (M ) → D b (M ) be the endo-functor with kernel E in D b (M × M ). Given a point m ∈ M , denote by Cm the corresponding sky-scraper sheaf. The following Lemma is used in [MM2]. Lemma 5.3. The homomorphism κ : Hom(Cm , Cm [1]) → Hom(E(Cm ), E(Cm )[1]), induced by E, is surjective, for all m ∈ M . Proof. The homomorphism κ ˜ : Hom(Cm , Cm [1]) → Hom(L(Cm ), L(Cm )[1]), induced by L, is an isomorphism, by Theorem 3.20. Set Em := E(Cm ) and Fm := L(Cm ). The exact triangle β

E [1] → F → O∆ → E [2] gives rise to the exact triangle βm

Cm [−1] → Em [1] → Fm → Cm . ∗ : Hom(F , F [1]) → Hom(E [1], F [1]) is surjective, by The pullback homomorphism βm m m m m an argument analogous to that in step 1 of the proof of Lemma 5.2. The kernel of the push-forward homomorphism βm,∗ : Hom(Em [1], Em [2]) → Hom(Em [1], Fm [1]) is the image of Hom(Em [1], Cm ) → Hom(Em [1], Em [2]). Now Hom(Em [1], Cm ) clearly vanishes. Hence, βm,∗ is injective. Given an element ξ ∈ Hom(Cm , Cm [1]), we have the equality βm ◦ E(ξ) = L(ξ) ◦ βm , since the homomorphism of kernels induces a natural transformation β : E[1] → L. We get the equality ∗ βm ◦κ ˜ = βm,∗ ◦ κ : Hom(Cm , Cm [1]) → Hom(Em [1], Fm [1]). ∗ ◦ κ The homomorphism βm ˜ is surjective, being a composition of such. Thus, βm,∗ ◦ κ is surjective. Hence βm,∗ is an isomorphism. Thus, κ is surjective. 

6. A deformation of the derived category D b (X) We prove Theorem 1.8 in this section. Although we talk of deforming categories, as will be seen presently, we shall only need to work with deformations of certain fixed objects and morphisms in a derived category. These are defined as follows. Definition 6.1. Let Y → S be a flat family of spaces (varieties or analytic spaces), and s a point of S. (1) A deformation of a perfect complex E ∈ D b (Ys ) is an S-perfect complex E ∈ D b (Y ) ∼ = together with an isomorphism ϕ : i∗ E −→ E , where i : Ys → Y is the closed immersion. (2) Given a morphism f : E1 → E2 between perfect complexes and deformations (E i , ϕi ), a compatible deformation of f is a morphism f : E 1 → E 2 such that f ◦ ϕ1 = ϕ2 ◦ i∗ f .

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49

6.1. Deformability of F . Denote by MΛ the moduli space parametrizing marked pairs (M, η) consisting of a holomorphic symplectic manifold M , and an isometry η : H 2 (M, Z) → Λ. fΛ → MΛ be the forgetful map given The moduli space MΛ is constructed in [Hu2]. Let φ : M f0 be the connected component of M fΛ in Theorem 1.8. Let by (M, η, A) 7→ (M, η). Let M Λ 0 0 0 f ). Then φ : M f → M0 is surjective, by MΛ be the connected component containing φ(M Λ Λ Λ [MM2, Theorem 4.14]. The following is the key result from [MM2] for the sequel; it is a direct consequence of Theorems 1.9(1), and 1.11(1), and Lemma 5.3 in that article. Theorem 6.2. Let X be a K3 surface with trivial Picard group. Then, the fiber of the f0 → M0 over (X [n] , η) consists of the single point (X [n] , η, A) ∈ M f0 , where morphism φ : M Λ Λ Λ A or A∗ is the modular Azumaya algebra of the Hilbert scheme X [n] (Def. 1.5). The following is an immediate corollary of Theorems 1.7 and 6.2 and the density of Hilbert schemes in M0Λ [MM1]. f0 , given in Equation (1.11), consisting of triples (X [n] , η, A) Corollary 6.3. The subset Hilb of M Λ f0 . For each such triple, A or where X is a K3 surface with a trivial Pic(X), is dense in M Λ A∗ is the modular Azumaya algebra over X [n] × X [n] .

f0 := Let M0 be the universal family7 over the moduli space of marked pairs M0Λ . Set M fΛ . There exists a universal Azumaya algebra A over M f0 (see [MM2, f0 × e 0 M M0 ×M0 M MΛ Λ Section 4]). f0 , containing Hilb, a Proposition 6.4. There exists a Zariski dense open subset U ⊂ M Λ universal twisted sheaf E over   f0 × e 0 U, f0 × e 0 M M2 := M M M Λ

Λ

satisfying E nd(E ) ∼ = A, and an extension (6.1)

E [1] → F → O∆M

f0 × e 0 U , and ∆M is the image of M via the of twisted sheaves over M2 , where M := M M Λ

diagonal embedding ∆ : M → M2 . This extension is non-split along every fiber of the e : M3 → U be the third fiber product of M over U and projection Π : M2 → U . Let Π 3 2 Πij : M → M , 1 ≤ i < j ≤ 3, the natural projections. The Brauer class Θ of E satisfies the equality (6.2)

Π∗12 (Θ)Π∗23 (Θ) = Π∗13 (Θ).

Consequently, both F and the convolution F ◦ F are objects of D b (M2 , Θ). Proof. Step 1: We show first that the Brauer class of A restricts as a trivial class to the f0 e 0 M f0 over a Zariski dense open subset of M f0 . Let µ2n−2 ⊂ C∗ be diagonal ∆M f0 ⊂ M ×M Λ Λ the group of roots of unity of order dividing 2n − 2. The exponential map   2πi(•) : Z → µ2n−2 exp 2n − 2 7While a universal family need not exist over the moduli space of marked pairs of a general class of holo-

morphic symplectic manifolds, such a family does exist over M0Λ as manifolds of K3[n] -type have no nontrivial automorphisms which act trivially on cohomology in degree 2. This follows for Hilbert schemes by a result of Beauville [Be2], and consequently also for their deformations by [HaT1, Sec. 2].

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E. MARKMAN AND S. MEHROTRA

factors through an isomorphism Z/(2n − 2)Z ∼ = µ2n−2 . Given a marked pair (M, η) in M0Λ we get an isomorphism η¯ : H 2 (M × M, µ2n−2 )f →[Λ/(2n − 2)Λ]2 f0 comes with a coset induced by the marking η. The component M Λ θ˜ = (θ˜1 , θ˜2 ),

(6.3)

with θ˜1 = −θ˜2 and θ˜i of order 2n−2 in Λ/(2n−2)Λ, such that the Brauer class of the Azumaya ˜ in H 2 (M × M, O ∗ ), Algebra A of a triple (M, η, A) in this component is the image of η¯−1 (θ) f0 , O ∗ ), having f0 × e 0 M by construction [MM2, Eq. (7.6)]. The Brauer class Θ of A in H 2 (M MΛ f0 , µ2n−2 ). The restriction ˜ in H 2 (M f0 × e 0 M order 2n − 2, is the image of a topological class Θ MΛ

˜ ˜ to the fibers M × M of the family over a marked pair (M, η) is η¯−1 (θ). of Θ ′ 0 f , over which the differential fibration π : M f0 f0 → M Given an open analytic subset U of M Λ Λ ˜ of the Brauer class restricts as a topologically trivial fibration, we get that the pullback ∆∗ Θ f0 is trivial over U ′ , by the vanishing of θ˜1 + θ˜2 mentioned above. to M ˇ 2-cocycle θ of O ∗f0 There exists a Θ-twisted sheaf E representing A for some Cech f0 M ×M f0 M

U ′,

Λ

[MM2, Section 2]. Over the above open subset E restricts to the diagonal with a trivial Brauer class. Hence, we may adjust the pull-back ∆∗ (Θ) by a 2-coboundary, and adjust the gluing of ∆∗ E over U ′ to get an untwisted sheaf W . For every triple (X [n] , η, A) in U ′ , where Pic(X) is trivial, the restriction of A to X [n] × X [n] is the modular Azumaya algebra or its dual, by Theorem 6.2, and so W |X [n] is isomorphic to L ⊗ [(∧2 T ∗ X [n] )/(OX [n] · σ)], for some line-bundle L, by Proposition 4.2 (use the isomorphism E ∗ ∼ = τ ∗ E of Lemma 4.3 for the dual 2 ∗ of the modular Azumaya algebra). Set W := ∧ Tπ /(L0 π ∗ R0 π∗ ∧2 Tπ∗ ). We conclude the isomorphism E nd(W ) ∼ = E nd(∆∗ E ),

(6.4)

of Azumaya algebras over a Zariski dense open subset of U ′ , by the density of Hilbert schemes of K3 surfaces with trivial Picard group (Corollary 6.3) and the upper-semi-continuity theorem. f0 . Hence, the isomorphism Both sides of the isomorphism (6.4) are defined globally over M ′′ 0 f containing the dense subset Hilb given in holds over a Zariski dense open subset U of M Λ Equation (1.11). The sheaf ∆∗ E is ∆∗ Θ-twisted. The left hand side of (6.4) is an Azumaya algebra with a trivial Brauer class. The isomorphism (6.4) implies that the cocycle ∆∗ Θ is a coboundary over U ′′ , ∆∗ Θ = δ(ζ). f0 × e 0 M f0 → M f0 be the projection to the first factor. After a refinement Step 2: Let π1 : M MΛ f0 we may change the gluing of E over U ′′ via the cochain f0 × e 0 M of the open covering of M MΛ

π1∗ (ζ), so that E is a twisted sheaf with respect to a new cocycle, denoted again by Θ, which restricts to the trivial cocycle along the diagonal (with value 1 along every triple intersection). Then ∆∗ E is an untwisted coherent sheaf, which is isomorphic to W ⊗ L , for some line bundle f0 . L over the subset U ′′ of M Λ f0 to U ′′ . Denote Step 3: Let π : M → U ′′ be the restriction of the universal family from M Λ ′′ by Π : M ×U ′′ M → U the projection from the fiber square. We show next that the relative homomorphism sheaf (6.5)

Hj := H omΠ (O∆M , E [j])

INTEGRAL TRANSFORMS AND DEFORMATIONS

51

is a line bundle over a Zariski dense open subset U of U ′′ containing Hilb for j = 2, and it f0 . Local vanishes over U for j = 0, 1. This U would be also Zariski dense and open in M Λ Grothendieck-Verdier duality yields the isomorphism h i  RH om R∆M∗ OM , E [j] ∼ = R∆∗ RH om(OM , ∆! E [j]) .

Applying the functor R(Π)∗ on both sides, using the vanishing of Ri ∆M∗ OM for i > 0, we get the isomorphism (6.6) RH omΠ (O∆ , E [j]) ∼ = Rπ∗ (ω ∗ ⊗ L∆∗ E [j − 2n]). π

M

M

The relative dualizing sheaf ωπ of the morphism π is trivial along each fiber and is hence the pullback of a line bundle over U ′′ . Thus, the sheaf H omΠ (O∆M , E [j]) is isomorphic to the 0-th direct image of the complex L∆∗M E [j − 2n], tensored by a line bundle. For x ∈ U ′′ , denote by E |x the restriction of E to the Cartesian square of the fiber over x. The function φj : U ′′ → Z given by x 7→ dim(H0 (Mx , L∆∗Mx E |x [j − 2n]))

is upper semi-continuous by [Hart, Prop. 6.4]. The value of φ2 on triples in Hilb is 1, by the calculation of the torsion sheaf T or2n−2 (E , ∆∗ OX [n] ) in Proposition 4.2. The set Hilb f0 , by Corollary 6.3. It follows that φ ≡ 1 on a dense open subset U2 of U ′′ . is dense in M Λ The sheaf H omΠ (O∆M , E [2]) is a line bundle over U2 , since U ′′ is integral. The vanishing of T or2n−j (E , ∆∗ OX [n] ) in Proposition 4.2, for j < 2, implies the claimed vanishing of Hj over a Zariski dense subset Uj of U ′′ containing Hilb, for j = 0, 1. Set U := U0 ∩ U1 ∩ U2 . Step 4: Set H := H2 , where H2 is the line-bundle given in Equation (6.5). The vanishing of Hj , for j = 0, 1, and a standard spectral sequence argument  yield an isomorphism ∗ −1 0 ∗ −1 ∼ Hom(O∆M , Π (H ) ⊗ E [2]) = H H omΠ (O∆M , Π (H ) ⊗ E [2]) . The right hand space is H 0 (U, OU ), by the projection formula and the definition of H . Choosing the constant section 1 of the latter we get a tautological extension Π∗ (H −1 ) ⊗ E [1] → F → O∆M .

Replacing the twisted sheaf E by π ∗ (H −1 ) ⊗ E we get the desired extension (6.1). Step 5: We prove in this step the equality (6.2). Set r := 2n − 2. Let F p H k (M2 , µr ) p,q be the decreasing Leray filtration associated to the morphism Π : M2 → U . Set E∞ := F p H p+q (M2 , µr )/F p+1 H p+q (M2 , µr ). We have the Leray spectral sequence converging to p,q E∞ with E2 terms of the form E2p,q := H p (U, Rq Π∗ µr ) and differential d2 : E2p,q → E2p+2,q−1 . The sheaf R0 Π∗ µr is the trivial local system µr and the sheaf R1 Π∗ µr vanishes, since the fibers of Π are simply connected. The sheaf R2 Π∗ µr is the direct sum [Λ/rΛ]⊕2 of two copies of the trivial local system [Λ/rΛ], since the markings provide such a trivialization. We conclude the following: E 2,0 = E 2,0 ∼ = H 2 (U, µr ) ∞

2

1,1 E∞ = 0,

E30,2 = E20,2 ∼ = [Λ/rΛ]⊕2 , E33,0 = E23,0 ∼ = H 3 (U, µr ), h i 0,2 = ker d3 : E30,2 → E33,0 . E∞

2,0 The description of E∞ implies that the homomorphism Π∗ : H 2 (U, µr ) → H 2 (M2 , µr ) is injective. The analogous description of the graded summands of the Leray filtrations of H 2 (Mn , µr )

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E. MARKMAN AND S. MEHROTRA

0,2 holds for the n-th fiber self-products Mn , n ≥ 1, where the E∞ is naturally a subgroup of ⊕n [Λ/rΛ] . We have seen in Step 2 that Θ belongs to the kernel of ∆∗ : H 2 (M2 , O ∗ ) → H 2 (M, O ∗ ). e ∈ H 2 (M2 , µr ) of Θ. Next we normalize We have seen, in addition, that there exists a lift Θ this lift so that it restricts trivially to the diagonal. The composition

∗ H 2 (M, µr ) → H 2 (M, µr )/F 1 H 2 (M, µr ) ֒→ H 0 (U, R2 π∗ µr ) → H 0 (U, R2 π∗ OM ) ∗ e to the trivial class. The rightmost homomorphism is injective, since the maps the class ∆ (Θ) e Picard group of a generic fiber of π : M → U is trivial. Consequently, the image of ∆∗ (Θ) 2,0 e belongs to E∞ with respect to the Leray filtration in H 0 (U, R2 π∗ µr ) is trivial and ∆∗ (Θ) 2 e = π ∗ (α). The of H (M, µr ). Hence, there exists a class α in H 2 (U, µr ), such that ∆∗ (Θ) e The image of Π∗ (α) image of π ∗ (α) in H 2 (M, O ∗ ) is trivial, since such is the image of ∆∗ (Θ). 2 2 ∗ e ∗ (α−1 ) in H (M , O ) is trivial as well, since Π factors through π. Let β be the class ΘΠ in H 2 (M2 , µr ). Then ∆∗ (β) is trivial in H 2 (M, µr ) and β maps to the Brauer class Θ in H 2 (M2 , O ∗ ). 2,0 The E∞ graded summands of the Leray filtrations of H 2 (Mn , µr ), n ≥ 1, are all equal to 1,1 H 2 (U, µr ) and the E∞ terms all vanish. Hence, the kernel of ∆∗ : H 2 (M2 , µr ) → H 2 (M, µr ) 0,2 maps injectively into the quotient E∞ , which is naturally a subgroup of [Λ/rΛ]⊕2 . Classes in the kernel map to classes of the form (−λ, λ). Choose a class λ ∈ Λ/rΛ, such that β maps to (−λ, λ). Similarly, the kernel of the pullback H 2 (M3 , µr ) → H 2 (M, µr ), via the diagonal embedding, maps injectively into [Λ/rΛ]⊕3 . The class Π∗12 (β)Π∗23 (β)Π∗13 (β −1 ) restricts trivially to the diagonal and maps to the class

(−λ, λ, 0) + (0, −λ, λ) + (λ, 0, −λ) = (0, 0, 0)

in Hence, the class Π∗12 (β)Π∗23 (β)Π∗13 (β −1 ) is trivial in H 2 (M3 , µr ). The latter class maps to the class Π∗12 (Θ)Π∗23 (Θ)Π∗13 (Θ−1 ) in H 2 (M3 , O)∗ . Equality (6.2) follows. This completes the proof of Proposition 6.4.  [Λ/rΛ]⊕3 .

Remark 6.5. The description of the graded pieces of the Leray filtration of H 2 (M2 , µr ) is Step 5 of the proof above applies to the restriction M2V of M2 to a contractible open subset V of U . 2,0 ∼ 3,0 In that case we get that E∞ = H 2 (V, µr ) vanishes as well, so does E∞ , and H 2 (M2V , µr ) = 0,2 ∼ e V of Θ e to E∞ = [Λ/rΛ]⊕2 . Similarly, H 2 (MV , µr ) ∼ = Λ/rΛ. Consequently, the restriction Θ ˜ for some class θ˜ in H 2 (MV , µr ). Part 2 of Theorem 1.8 follows. M2V is equal to π1∗ (θ˜−1 )π2∗ (θ) Let X be a K3 surface, M := X [n] , and δ : F → F ◦F the comultiplication of the comonad object (1.10). Denote by πij : M × M × M → M × M the projection onto the ij-th factor. ∗ ⊣π The adjunction π13 13∗ yields the second isomorphism below. Hom(F , F ◦ F ) = Hom(F , π13 [π ∗ F ⊗ π ∗ F ]) ∼ = Hom(π ∗ F , π ∗ F ⊗ π ∗ F ). ∗

12

23

13

12

23

∗ F → π ∗ F ⊗ π ∗ F . We get the natural morphism Denote the image of δ by δ˜ : π13 23 12 ∗ ∼ ˜ π13 (δ) : π13 π F = F ⊗C Y (OM ) −→ F ◦ F , ∗



13

˜ with the morphism where Y (OM ) ∈ is the Yoneda algebra of M . Composing π13∗ (δ) 1F ⊗ ι : F ⊗C λn → F ⊗C Y (OM ) (see (2.5)) we get the natural morphism D b (pt)

(6.7)

m : F ⊗ λn → F ◦ F .

Lemma 6.6. The morphism m, given in Equation (6.7), is an isomorphism, in the case where M is the Hilbert scheme X [n] , the universal sheaf U ∈ D b (X × X [n] ) is the ideal sheaf of the universal subscheme, and F is the modular complex U ◦ U ∨ [2].

INTEGRAL TRANSFORMS AND DEFORMATIONS

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Proof. The morphism m is obtained from the sequence of morphisms (2.5) η⊗id

ι

m

∆∗ OX ⊗C λn −→ ∆∗ OX ⊗C Y (OM ) −→ A ⊗C Y (OM ) −→ A .

by pre-convolution with U ∨ [2], and post-convolution with U . Here we use the fact discussed at the end of Construction 2.1 that for a morphism f : T → S, the monadic action of f∗ f ∗ on objects of the form f∗ (G ) is compatible with the action of the algebra f∗ (OT ). The statement then follows immediately from Theorem 2.2, Part (1), which says that the composition of the morphisms in the display above is an isomorphism.  Let M and U be as in Theorem 1.8 and let Mn denote the n-th fiber self-product of M over U . The convolution F ◦ F is defined relative to U , i.e., the tensor product is taken over M3 . Let Π : M2 → U be the natural morphism. Lemma 6.7. (1) The relative endomorphism sheaf H omΠ (F , F ) is canonically isomorphic to the structure sheaf over a dense open subset of U containing the locus Hilb of Corollary 6.3. (2) The relative homomorphism sheaves  H omΠ (F , F ◦ F ), and H omΠ (F , F ◦ F ◦ F ) are both line bundles on a dense open subset of U . Proof. We shall consider only the sheaf H omΠ (F , F ◦ F ) of part (2). The other sheaves are shown to be line-bundles in the same way, and the sheaf H omΠ (F , F ) clearly has a global non-vanishing identity section establishing its triviality. For y := (M, η, A) ∈ U , denote the restriction F |M ×M more simply as F y . When y belongs to Hilb, i.e., when M = X [n] where X is a K3 surface with trivial Pic(X), we have Fy ◦ Fy ∼ = F y ⊕ F y [−2] ⊕ · · · ⊕ F y [2 − 2n], by Theorem 6.2 and Lemma 6.6 (note that when A is the dual of the modular Azumaya algebra F y is the pullback of the modular complex by the automorphism τ of M × M interchanging the two factors, by lemma 4.3). Furthermore, (F ◦ F )y is isomorphic to F y ◦ F y , by the base change theorem [Hart, Proposition 6.3]. The locus U ′ where the fiber-wise homorphisms Hom(F y , F y ◦F y ) are one-dimensional is locally closed by semi-continuity [Hart, Proposition 6.4]. U ′ contains every y ∈ Hilb, since Hom(F y , F y [k]) vanishes for the modular F y of a Hilbert scheme and for k < 0, by Lemma 5.1. U ′ is a dense open subset of U , by Corollary 6.3.  We will denote again by U the open subset where the statement of Lemma 6.7 holds. Similarly, we will continue to denote the universal family by π : M → U and its fiber square by Π : M2 → U . Denote the restriction to D b (M2 , Θ) of the extension constructed in Proposition 6.4 by the same symbol F . To define the structure of a monad object on F extending the modular one on the Hilbert schemes, we need to produce a counit ǫ, and a comultiplication δ. The former structure map is in fact given by the very definition of F in triangle (6.1): ǫ

E [1] −→ F −→ O∆M .

We shall presently produce the other, and verify that the structure maps satisfy the necessary compatibilities.

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Any extension δ : F → F ◦ F of the comultiplication is required to give a section of the two morphisms (6.8)

F ◦ ǫ : F ◦ F −→F , and, ǫ ◦ F : F ◦ F −→F ,

that is, (F ◦ ǫ) ◦ δ = (ǫ ◦ F ) ◦ δ = 1F . Conversely, the morphisms (6.8) admit sections over a dense open subset of U extending the modular comultiplication; we shall call them δ 1 and δ 2 , respectively. Indeed, consider the following pair of morphisms of sheaves over U obtained from (6.8): F ◦ǫ

H omΠ (F , F ◦ F )

ǫ◦F

/

= OU / H omΠ (F , F ) ∼

These morphisms are isomorphisms over a dense open subset U 0 of U containing Hilb, by Corollary 6.3 and Lemma 5.1. Define δ 1 and δ 2 to be the pre-images of 1F . Proof of Theorem 1.8 (1). First suppose that there exists a comultiplication δx for F |x at some point x ∈ U 0 . Then, δ 1 |x = δ2 |x = δx since δx is the unique section of both morphisms in Equation (6.8), by the discussion above. To complete the proof, first note that the equality δ 1 = δ 2 holds over U 0 by the fact that we have established it over a dense subset of it. Second, note that the coassociativity (1.6) and counit laws (1.7) amount to equalities of two sections of the line bundles H omΠ (F , F ◦ F ), respectively H omΠ (F , F ◦ F ◦ F ), over U 0 . These equalities hold, again because they have been established for a dense subset of U 0 . Finally we rename U 0 as U .  Remark 6.8. Let Π13 : M3 → M2 be the projection on the first and third factors. Let λn be the object fitting in the exact triangle (6.9)

λn → RΠ13∗ OM3 → R2n Π13∗ OM3

in D b (M2 ), where the right morphism is well defined due to the vanishing of Ri Π13∗ OM3 for i > 2n. The construction of the comultiplication δ : F → F ◦ F enables us to extend the isomorphism m in Equation 6.7 to a morphism (6.10)

m : F ⊗OU λn → F ◦ F

of objects over M2 , which is an isomorphism over a dense open subset W of U containing Hilb, by Lemma 6.6 and the argument in the proof of Lemma 2.3. Again we rename W as U .

Lemma 6.9. Let w := (M, η, A) be a point of the open set W of Remark 6.8, where M ∼ = MH (v) is isomorphic to a moduli space of H-stable sheaves over some K3 surfaces X, E w is the modular sheaf over M × M associated to a twisted universal sheaf U over X × MH (v), and A is the modular Azumaya algebra E nd(E w ). Let A be the monad object over X × X associated to U . Then the morphism α : ∆∗ OX ⊗C λn → A , given in Equation (2.5), is an isomorphism as well.

Proof. This follows from the fact that mw is the image of α via the functor Γ, given in (5.2), and the composition Γ†R ◦ Γ of Γ with its right adjoint admits the identity endo-functor of D b (X × X) as a direct summand, by Theorem 2.2 (2). Denote by Σ the endofunctor of D b (X × X), such that Γ†R ◦ Γ ∼ = 1Db (X×X) ⊕ Σ. We have the equalities α ⊕ Σ(α) = (Γ†R ◦ Γ)(α) = Γ†R (mw )

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55

Then α ⊕ Σ(α) is an isomorphism, since mw is. Hence, each of α and Σ(α) must be an isomorphism as well.  6.2. The triangulated structure on the category of comodules. We give a proof of part 3 of Theorem 1.8 here, namely, that the category D b (MV , θ)L constructed above carries a 2-triangulated structure. This amounts to verifying Balmer’s separability criterion [Bal1] for the comonad L. We start by briefly recalling some of the necessary background. Let Ψ : C → D be a functor with left adjoint Φ. We say that Ψ is separable if the unit η : 1D → ΨΦ has a natural retraction8 ξ : ΨΦ → 1C . An additive category C is said to be suspended if it is equipped with an auto-equivalence, called suspension, [1] : C → C (which, for simplicity, is considered an isomorphism, [1]−1 [1] = 1C = [1][1]−1 ). For example, any triangulated category is a suspended category. A functor between suspended categories is called a suspended functor if it commutes with suspensions. In general, a property P of suspended categories or functors is stably P if its definition respects the suspended structures involved. Thus, a suspended functor Φ : D → C is stably separable if it is separable, and the splitting ξ commutes with suspension. Let C be a category, and Λ = hΛ, ǫ, δi a comonad on C. Denote by C the category of comodules of Λ. The forgetful functor F : C → C has a right adjoint, the free comodule functor, G : C → C , which is defined as G(a) := (Λa, δa ). It is easy to see that if C and Λ are suspended, then so is C , and the pair of functors F and G commute with suspension. Definition 6.10. A comonad Λ on a category C is said to be a separable comonad if there exists a natural retraction δˆ : Λ2 → Λ of the comultiplication δ : Λ → Λ2 such that (6.11)

ˆ ◦ Λδ = δ ◦ δˆ = Λδˆ ◦ δΛ δΛ

[Bal1, Def. 3.5]. If C is suspended, Λ is said to be stably separable if the various functors and natural morphisms in question respect the suspension. The following abridged form of Balmer’s Main Theorem [Bal1] is all we need for our purposes here: Theorem 6.11 ([Bal1], Theorem 5.17). Let C be an idempotent-complete category with a triangulation of order N ≥ 2, and let Λ be a stably separable co-monad on C, such that Λ : C → C is exact up to order N . Then, the category of Λ-comodules C admits a triangulation of order N such that, both the free comodule functor G : C → C and the forgetful functor F : C → C are exact up to order N . In fact, each of these properties characterizes the triangulation on the category C . Remark 6.12. Rather than go into the specifics of N -triangulations, we simply observe what is relevant for us (see [Bal1, Section 5]): A 2-triangulated category satisfies all the axioms of a triangulated category, except the octahedral axiom, while a 3-triangulated category is also a triangulated category in the sense of Verdier (but not vice-versa). In our intended application of the above result, C will be D b (MV , θ)L , the derived category of an abelian category, which, as such, is in fact canonically N -triangulable for all N ≥ 2 [Malt, Corollaire, p. 18]. While the comonad L is certainly 2-exact, it is not clear at all that it is N -exact for this structure when N > 2. We expect this is to be true, at least over algebraic fibers Mu of MV , but are unable to prove this at the moment. 8Recall that ξ is a retraction of η if it is a left inverse of η, i.e., if ξη : 1 → 1 is the identity natural D D

transformation. In this case η is a section of ξ.

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Lemma 6.13. Let Λ := (Λ, ǫ, δ) be a stable comonad on a suspended category C realized by a stable adjunction Φ : D ⇄ C : Ψ. If the functor Ψ is stably separable, then the comonad Λ is stably separable. Proof. Note first that Λ := ΦΨ, ǫ : Λ → 1C is the counit, and δ = ΦηΨ. The statement essentially follows from the proof in [BBW, §2.9]. We recall the easy details for the sake of completeness: Given a retraction ξ of η : 1D → ΨΦ, we obtain a retraction of δ = ΦηΨ by ˆ ◦ Λδ = δ ◦ δ, ˆ as the other equality in setting δˆ = ΦξΨ. We will prove only the condition δΛ (6.11) follows in exactly the same way. ˆ ◦ Λδ = δ ◦ δˆ translates to (ΦξΨΦΨ) ◦ (ΦΨΦηΨ) = (ΦηΨ) ◦ (ΦξΨ), which in The equality δΛ turn would follow from (ξΨΦ) ◦ (ΨΦη) = η ◦ ξ. The latter states, for every object x of D, the commutativity of the diagram: (ΨΦ)(x)

(ΨΦ)ηx

/ (ΨΦ)(ΨΦ)(x) (ξ(ΨΦ))x

ξx ηx



x



/ (ΨΦ)(x),

which follows from the naturality of ξ for the arrows ηx .



Proof of Theorem 1.8(3). Fix a contractible Stein open subset V of U as in the statement of the Theorem. We shall work over V , but will continue to use the same notation as above for the restrictions of the various morphisms appearing in parts (1), and (2) of this result. To show that D b (MV , θ)L carries a natural 2-triangulated structure as in the statement, it suffices to produce a retraction δˆ : F V ◦F V → F V of the comultiplication δ : F V → F V ◦F V by Theorem 6.11, such that the following two diagrams commute: δˆ F ◦ F ❴ ❴ ❴ ❴ ❴/ F

(6.12)



δ◦F



δˆ F ◦ F ❴ ❴ ❴ ❴ ❴/ F

δ



F ◦ F ◦ F ❴ ❴ ❴/ F ◦ F

F ◦δ



δ

F ◦ F ◦ F ❴ ❴ ❴/ F ◦ F

F ◦δˆ

ˆ δ◦F

Consider the triangle (6.9). The object RΠ13∗ OM3 is canonically isomorphic to Π∗ Rπ∗ OMV V by base-change, where π and Π are the structure morphisms MV → V and M2V → V , respectively. As all extensions of line bundles vanish over V , this object is canonically split: RΠ13∗ OM3 ∼ = OM2 ⊗OV Y (OMV ), where Y (OMV ) := ⊕ni=0 R2i π∗ OMV . Define δˆ to be the V

V

m−1

composition F V ◦ F V → F V ⊗OV λn → F V , where m is given in Equation (6.10) and the ˆ second arrow is induced by the splitting above. We first observe that with this definition of δ, the diagrams (6.12) commute when restricted to points corresponding to Hilbert schemes with their modular complexes. Indeed, in this case, the morphism m is nothing but the morphism on kernels corresponding to the following retraction of the comultiplication ΦξΨ

ΦΨΦΨ = Λ2 −→ ΦΨ = Λ.

Here (Φ, Ψ) is the adjoint pair Φ : D b (X) ⇄ D b (X [n] ) : Ψ realizing our comonad Λ, and ξ : ΨΦ → 1Db (X) is the retraction of η : 1Db (X) → ΨΦ given by Thoerem 2.2, Part (1). Thus (the restrictions of) the two diagrams (6.12) commute by Lemma 6.13.

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57

Consider the sheaf H omΠ (F ◦ F , F ◦ F ). It follows from Lemma 5.1, and semi-continuity [Hart, Proposition 6.4] that this sheaf is a vector bundle on a dense open subset of U containing Hilb. As above, we denote this set by U also. The latter is the open subset of Theorem 1.8. ˆ ◦ (δ ◦ F V ); we note that dx = 0 at points x ∈ Hilb ∩ V . Therefore, Set d = δ ◦ δˆ − (F V ◦ δ) since this locus is dense, d = 0 on V . The commutativity of the left diagram above follows. The argument establishing the commutativity of the right diagram is similar.  6.3. Monodromy invariance. We prove part 5 of Theorem 1.8 in this section. We first f0 and observe that the open subset U in recall the monodromy action on the moduli space M Λ Theorem 1.8 may be chosen to be monodromy invariant. We then use a density theorem of Verbitsky to deduce the stated property of U . The isometry group O(Λ) acts on the moduli space of marked pairs MΛ as follows. An element g ∈ O(Λ) acts on a marked pair (M, η) by g(M, η) = (M, gη). Let M0Λ be a connected component of MΛ of marked pairs of K3[n] -type. Denote by G ⊂ O(Λ) the subgroup which send M0Λ to itself. The subgroup G is related to the monodromy subgroup M on2 (M ) of the isometry group of the second integral cohomology a manifold M of K3[n] -type as follows. Given a pair (M, η) in M0Λ , we have the equality G = η ◦ M on2 (M ) ◦ η −1 . Given an element u ∈ Λ satisfying (u, u) = ±2, let Ru be the reflection given by Ru (x) := x − 2(u,x) (u,u) u. Set  Ru if (u, u) = −2 ρu := −Ru if (u, u) = 2.

The group G is the subgroup of O(Λ) generated by {ρu : (u, u) = ±2}, by [Ma4, Theorem 1.2]. There exists a character cov : G → Z/2Z  0 if (u, u) = −2 satisfying cov(ρu ) = (see [Ma3, Sec. 4.1]). The unordered pair of cosets 1 if (u, u) = 2. {θ˜1 , −θ˜1 } in Λ/(2n − 2)Λ, given in Equation (6.3), is G-invariant and g(θ˜1 ) = (−1)cov(g) θ˜1 , by f0 be a connected component of M fΛ containing a triple (X [n] , η0 , A0 ), [Ma5, Lemma 7.3]. Let M 0 Λ [n] [n] where A0 is the modular Azumaya algebra over the cartesian square X0 × X0 of the Hilbert scheme of a K3 surface X0 . Theorem 6.14.

fΛ , given by (1) The G-action on M

cov(g) )

g(M, η, A) = (M, gη, A(∗

)

f0 to itself, where A(∗cov(g) ) is A, if cov(g) = 0, and maps the connected component M Λ A∗ , if cov(g) = 1. f0 , in Theorem 1.8, may be chosen to be invariant with respect (2) The open subset U of M Λ to the above action of G. Proof. 1) This statement is a version of [MM2, Theorem 1.11] and its proof is included in the proof of that Theorem. 2) All the properties that points of U were required to satisfy depend on the isomorphism class of the Azumaya algebra. Hence, U may be enlarged replacing it by the union of all translates g(U ), for all g in the kernel of cov. We may thus assume that U is ker(cov)invariant.

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Choose an element g ∈ G with cov(g) = 1. The dense subset Hilb is G-invariant, by definition. Hence, Hilb is contained in U ∩ g(U ). The latter is G-invariant.  Example 6.15. Let D ⊂ X [n] be the divisor of non-reduced subschemes, d ∈ H 2 (X [n] , Z) the class of D, and Rd : H 2 (X [n] , Z) → H 2 (X [n] , Z) the reflection given by Rd (x) = x − 2(x,d) (d,d) . Then Rd is a monodromy operator, by [Ma3, Cor. 1.8], and a Hodge isometry. We have cov(Rd ) = 1, by [Ma3, Lemma 4.10(4)] (Rd is the image of the duality operator v 7→ v ∨ via the homomorphism f in that Lemma). Let A be the modular Azumaya algebra over X [n] ×X [n] and η a marking for X [n] . Then the triples (X [n] , η, A) and (X [n] , ηRd , A∗ ) belong to the same f0 and have the same period. Hence, the two triples are inseparable connected component M Λ points in moduli. f0 , by Theorem Proof of part 5 of Theorem 1.8. U is a non-empty open G-invariant subset of M Λ 0 6.14. Hence, its image φ(U ) in MΛ is a G-invariant non-empty open subset. The main result of [V4, Theorem 4.11] states that the G-orbit of a marked hyperk¨ahler manifold (M, η) with second Betti number b2 (M ) ≥ 5 and Picard rank ≤ b2 (M ) − 3 is dense in its connected component M0Λ . Hence, φ(U ) contains (M, η), for every marking η, such that (M, η) belongs f0 in the moduli space of marked pairs. to the image M0Λ of M  Λ The following is a sufficient condition for a triple to belong to the open subset U mentioned in Theorem 1.8.

f0 is dense in M f0 , and is thus contained Lemma 6.16. The G-orbit of a point (M, η, A) of M Λ Λ in the open subset U of Theorem 1.8 supporting the deformation hF , ǫ, δi of comonad objects, as well as in the open subset W of Remark 6.8 where the the square F ◦ F is isomorphic to a direct sum of shifts of F , provided the following conditions are satisfied. (1) The rank of Pic(M ) is ≤ 20. ˜ (2) The order of the Brauer class of A, which is the image in H 2 (M × M, O ∗ ) of η −1 (θ), ˜ is 2n − 2. Here θ is the class given in (6.3). Proof. Condition (1) implies that the fiber φ−1 (φ(M, η, A)) intersects U , by Theorem 1.8 (5). Condition (2) implies that the fiber φ−1 (φ(M, η, A)) consists of a single point. Indeed, the condition implies that the Azumaya algebra A′ of a triple (M, η, A′ ) in this fiber is slope stable with respect to every K¨ ahler class on M × M , by [Ma5, Prop. 7.8]. It follows that A is isomorphic to A′ , by [MM2, Lemma 5.3].  Remark 6.17. Let X be a K3 surface, v a primitive algebraic Mukai vector, and H a v-generic polarization, such that the dimension of M := MH (v) is ≥ 4. Denote by A the modular Azumaya algebra over M × M (Definition 1.5). There exists a marking η for M , such that f0 , if and only if A is π ∗ ω + π ∗ ω slope-stable, as an Azumaya algebra, (M, η, A) belongs to M 2 1 Λ with respect to some K¨ ahler class ω on M . The above slope-stability of A with respect to every K¨ ahler class on M is known when the order of the Brauer class of A is equal to the rank 2n − 2 of A, by [Ma5, Prop. 7.8], as well as when Pic(X) is trivial and v = (1, 0, 1 − n) is the Mukai vector of the ideal sheaf of a length n subscheme, so that M = X [n] , by [Ma7]. Stability of the modular Azumaya algebra over X [n] × X [n] with respect to some K¨ ahler class on X [n] is known whenever the rank of Pic(X) is less than 20, by [Ma7]. Stability being a f0 follows for the generic member of Zariski open condition, membership of (MH (v), η, A) in M Λ a family of such moduli spaces over an irreducible base, once known for some fiber.

INTEGRAL TRANSFORMS AND DEFORMATIONS

59

Corollary 6.18. Let X be a K3 surface of Picard rank ≤ 19, v a primitive algebraic Mukai vector, H a v-generic polarization, such that the dimension of MH (v) is ≥ 4. Assume further that (6.13)

gcd{(u, v) : u ∈ H ∗ (X, Z) and c1 (u) ∈ H 1,1 (X)} = (v, v).

Then the morphism α given in Equation (2.5) is an isomorphism (so the monad object A is totally split). Proof. The Picard rank of MH (v) is ≤ 20. The order of the Brauer class of the modular Azumaya algebra A over MH (v) × MH (v) is equal to the left hand side of Equation (6.13), by [Ma5, Lemma 7.3]. The rank of the Azumaya algebra A is equal to the right hand side of Equation (6.13). Thus A is slope-stable, as an Azumaya algebra, with respect to every K¨ aher class, by [Ma5, Prop. 7.8]. Hence, there exists a marking η, such that the triple (MH (v), η, A) corresponds to a point in the open set U of Theorem 1.8, by Lemma 6.16. The assertion now follows from Lemma 6.9.  6.4. A K3 category. Let M be an irreducible holomorphic symplectic manifold of K3[n] -type admitting a deformed comonad structure L := (L, ǫ, δ) constructed above. Proposition 6.19. The shift by [2] is a Serre functor for the category D b (M, θ)L over a dense f0 of triples. In other words, G-invariant open subset, containing Hilb, of the moduli space M Λ given objects a and b of D b (M, θ)L , there exists a natural isomorphism (6.14)

Hom(a, b)f →Hom(b, a[2])∗ .

ˆ : D b (M, θ) → D b (M, θ)L be the natural functor taking an object x of D b (M, θ) Proof. Let L to (L(x), δx : L(x) → L2 (x)). The full subcategory of D b (M, θ)L with objects of the form ˆ L(x), for an object x of D b (M, θ), will be denoted by D b (M, θ)Lf r . We consider first the ˆ ˆ and b = L(y) for objects x, y of D b (M ). Denote by case a, b ∈ D b (M, θ)Lf r , with a = L(x) ˆ is the right adjoint of F . The right F : D b (M, θ)L → D b (M, θ) the forgetful functor. Then L adjoint of the functor L is isomorphic to L[2n − 2] over the dense subset of the moduli space of triples consisting of Hilbert schemes, by Lemma 4.1. The kernel F of the functor L has a one dimensional space Hom(F , F ) if M is a Hilbert scheme, and so the set over which the isomorphism of Lemma 4.1 holds is a dense open subset. We omit the proof of the latter statement, which is similar to the proof of Lemma 6.7, followed by that of Lemma 2.3. Over this open set we get ˆ ˆ ˆ ˆ Hom(x, F L(y)[2n−2]) = Hom(x, L(y)[2n−2]) ∼ y) ∼ L(y)), = Hom(L(x), y) = Hom(F L(x), = Hom(L(x), ˆ = L and the isomorphisms are due where the equalities above follow from the equality F L ˆ ˆ to the adjunctions L ⊣ L[2n − 2] and F ⊣ L. We conclude that L[2n − 2] is a left adjoint b ′ b L to the restriction F : D (M, θ)f r → D (M, θ) of the forgetful functor F to the subcategory D b (M, θ)Lf r of D b (M, θ)L . We get the bi-functorial isomorphisms (6.15)

∗ ∼ ˆ ˆ ˆ ˆ Hom(L(x), L(y)) L(x)[2]) . = Hom(L(x), y) ∼ = Hom(y, L(x)[2n])∗ ∼ = Hom(L(y),

ˆ and the equality F L ˆ = L. The second is The first isomorphism is due to the adjunction F ⊣ L b ˆ Serre Duality for D (M, θ). The last is due to the adjunction L[2n−2] ⊣ F ′ . Thus, D b (M, θ)Lf r is a K3 category.

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∗ the isomorphism given in Equaˆ ˆ ˆ ˆ Denote by σx,y : Hom(L(x), L(y)) → Hom(L(y), L(x)[2]) ˆ ˆ tion (6.15). Let e : L(w) → L(x) be a morphism. Bi-functoriality yields the commutative diagram

ˆ ˆ Hom(L(x), L(y))

σx,y

∗ ˆ ˆ / Hom(L(y), L(x)[2]) (e∗ )∗

e∗



ˆ ˆ Hom(L(w), L(y))

σw,y



∗, / Hom(L(y), ˆ ˆ L(w)[2])

ˆ ˆ and the equality σw,y e∗ = (e∗ )∗ σx,y . Similarly, given a morphism f : L(y) → L(z), we get the analogous equality σx,z f∗ = (f ∗ )∗ σx,y ˆ ˆ ˆ ˆ of homomorphisms from Hom(L(x), L(y)) to Hom(L(z), L(x)[2]). If w = x and y = z the two equalities above yield the equality (6.16)

σx,y (e∗ f∗ ) = (e∗ )∗ (f ∗ )∗ σx,y .

The comonad category D b (M, θ)L is the idempotent completion of the category of free comodules D b (M, θ)Lf r . This follows from [Bal1, Prop. 2.10] and triangulated Barr-Beck ˆ [MS, E] (also see [Bal2]). Objects of the idempotent completion are pairs (L(x), e), where e ∈ ˆ ˆ ˆ ˆ Hom(L(x), L(x)) is an idempotent. A morphism in Hom((L(x), e), (L(y), f )) is a morphism ˆ ˆ ˆ ˆ g : L(x) → L(y) satisfying f g = g = ge. In other words, Hom((L(x), e), (L(y), f )) is the image ∗ ˆ ˆ of the idempotent endomorphism e f∗ of Hom(L(x), L(y)). Set e1 := e∗ f∗ , e2 := e∗ (1−f )∗ , e3 := (1−e)∗ f∗ , and e4 := (1−e)∗ (1−f )∗ . These are commutP ˆ ˆ ing idempotent endomorphisms of Hom(L(x), L(y)) satisfying ei ej = 0, if i 6= j, and 4i=1 ei = 1. Set e˜1 := (e∗ )∗ (f ∗ )∗ , e˜2 := (e∗ )∗ ((1−f )∗ )∗ , e˜3 := ((1−e)∗ )∗ (f ∗ )∗ , e˜4 := ((1−e)∗ )∗ ((1−f )∗ )∗ . ∗ satisfying e ˆ ˆ These are commuting idempotent endomorphisms of Hom(L(y), L(x)[2]) ˜i e˜j = 0, P4 P4 P4 if i 6= j, and i=1 e˜i = 1. We get the decomposition σx,y = i=1 j=1 e˜j σx,y ei . Equation (6.16) implies that σx,y ei = e˜i σx,y , 1 ≤ i ≤ 4 (note that the equation holds with e replaced by 1 − e or f replaced by 1 − f ). Hence, e˜j σx,y ei = 0, if i 6= j. Consequently, σx,y maps the image of ei isomorphically onto the image of e˜i , for 1 ≤ i ≤ 4. Considering the case i = 1 we ˆ ˆ conclude that σx,y maps Hom((L(x), e), (L(y), f )) = Im(e1 ) isomorphically onto Im(˜ e1 ). Now ∗ ˆ ˆ Im(˜ e1 ) maps isomorphically onto Hom((L(y), f ), (L(x)[2], e)) via the natural homomorphism ∗ → Hom((L(y), ˆ ˆ ˆ ˆ Hom(L(y), L(x)[2]) f ), (L(x)[2], e))∗ . We thus obtain the desired isomorphism (6.14).  7. Comparison with Toda’s Category The first order deformations of the category of coherent sheaves Coh(S) on a smooth, projective variety S are parametrized by its degree 2 Hochschild cohomology HH 2 (S). This has an interesting interpretation via the HKR-isomorphism, ∼ =

I ∗ : HT 2 (S) = H 0 (∧2 TS ) ⊕ H 1 (TS ) ⊕ H 2 (OS ) → HH 2 (S), namely, the general deformation may be understood as composed of non-commutative, complex and “gerby” parts corresponding to the three summands. Given a class η ∈ HT 2 (S), Toda gave an explicit construction ([To]) of the corresponding infinitesimal deformation Coh(S, η): Let η0,2 be the component of η in H 2 (OS ). Then Coh(S, η) is the C[ε]/(ε2 )-linear abelian category of η0,2 -twisted coherent sheaves of modules

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over a deformed “structure sheaf” OSη of noncommutative C[ε]/(ε2 )-algebras. Further, he proved that these deformations behave functorially under Fourier-Mukai transformations: Theorem 7.1 ([To]). Let S and Y be smooth, projective varieties, and suppose that there is ∼ a Fourier-Mukai equivalence Φ : D b (S) → D b (Y ). If φ : HH ∗ (S) → HH ∗ (Y ) is the induced map on cohomology, there is an equivalence Φ† : D b (Coh(S, η)) → D b (Coh(Y, φ(η))) such that the following diagram commutes (up to natural isomorphisms of functors): D b (S)

i∗

i∗

/ D b (Coh(S, η))

/ D − (S) Φ−

Φ†

Φ



D b (Y )

i∗



/ D b (Coh(Y, φ(η)))

i∗



/ D − (Y )

The notation D − refers to the derived category of bounded above complexes of coherent sheaves, while i∗ and i∗ stand for restriction and extension of scalars (see [To, Section 4]), respectively. In this section, M will denote a moduli space MH (v). Our goal is to prove that the category of comodules constructed in the previous section via deformations of M agrees infinitesimally with Toda’s category. In other words, for every direction in the 21-parameter space that F deforms, there is a class η ∈ HT 2 (X) for our K3 surface X such that the comodule category over the dual numbers is exact equivalent to D b (Coh(X, η)). This is Theorem 7.15. It is proven under the following assumption, which will remain in force sections §7.2 and §7.3. Assumption 7.2. The moduli space M = MH (v) is a fine moduli space on an algebraic K3 surface X, supporting a universal family U , such that the kernel U ∨ [2] ◦ U is totally-split (as f0 such that A is the modular in Definition 3.1). Further, there exists a triple (M, η, A) ∈ M Λ Azumaya algebra of the moduli space MH (v) (see Remark 6.17). Remark 7.3. The assumption of algebraicity on X is in order to use the results of [To], which are stated in the context of smooth and projective varieties. One expects these results to hold over proper analytic manifolds also, but this generalization does not appear in the literature. 7.1. Hochschild cohomology and deformations. Let Φ : D b (S) → D b (Y ) be a FourierMukai equivalence; write P ∈ D b (S × Y ) for its kernel and Q ∈ D b (Y × S) for that of its inverse. We have an isomorphism φ : HH i (S) → HH i (Y ) defined as (7.1)

η

P◦η◦Q

(O∆S −→ O∆S [i]) 7→ (O∆Y = P ◦ O∆S ◦ Q −→ P ◦ O∆S [i] ◦ Q = O∆Y [i])

In particular, for i = 2, this defines a bijective correspondence between the infinitesimal deformations of Coh(S) and those of Coh(Y ). Note, however, that this makes use of the fact that Φ is invertible. In fact, one does not have functoriality for Hochschild cohomology under general integral transforms. For later use, we state a criterion of Toda and Lowen for when an object in the derived category can be deformed to first order. First recall the construction of the Atiyah class a(G ) ∈ Hom(G , G ⊗ ΩS [1]) for any object G ∈ D b (S): Regarding the sequence 0 → I∆S /I∆2 S → OS×S /I∆2 S → O∆S → 0

as a sequence of Fourier-Mukai kernels, and taking integral transforms of G accordingly, we obtain the triangle (7.2)

G ⊗ ΩS → π2,∗ (π1∗ (G ) ⊗ OS×S /I∆2 S ) → G .

Then, a(G ) is the extension class of (7.2).

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Theorem 7.4. ([To, Prop. 6.1]; [Lo, Thm. 1.1]) Let G ∈ D b (S) and u ∈ HT 2 (S). There exists a perfect object G 1 ∈ D b (S, u) such that i∗ G 1 ∼ = G if and only if u · exp a(G ) = 0 in Hom(G , G [2]). P i Here exp a(G ) = a (G ), the summand ai (G ) ∈ Hom(G , G ⊗ Ωi [i]) being the i-fold composition of a(G ) with itself, followed by anti-symmetrization. Remark 7.5. The sufficiency of the vanishing of u·exp a(G ) for the deformability of G was first proven by Toda, op. cit.. The necessity of this condition in this criterion follows from Lowen’s work, who proved that the obstruction to the deformability of G is precisely the image of u under the characteristic morphism: χ

G HH 2 (S) −→ HomS (G , G [2]).

Regard the elements of HH 2 (S) as natural transformations between the functors 1Db (S) and [2]. Evaluating them on G defines χG . The statement of the criterion can then be deduced from the commutativity of the following diagram (see [C3, Proposition 4.5]): HT 2 (S) I∗



HH 2 (S)

· exp a(G )

/ HomS (G , G [2]). ❥❥5 ❥ ❥❥ ❥ ❥ ❥ ❥ ❥❥❥ χG ❥❥❥

Remark 7.6. Theorem 7.4 is the main component of the proof Theorem 7.1. Using it, [To] proves that that the kernel of the Fourier-Makai transform Φ : D b (S) → D b (Y ) deforms to the derived category of the first-order deformation of Coh(S × Y ) in the direction p∗S (−ˇ η) + p∗Y (φ(η)). Here (ˇ) denotes the action of transposition of factors on Hochschild cohomology. Example 7.7. The identity functor D b (S) → D b (S) is interesting in light of Theorem 7.1. Its η ) + p∗2 (η) by Remark 7.6. It is possible kernel O∆ must deform along every direction p∗1 (−ˇ to explicitly define a canonical deformation, the “structure sheaf of the deformed diagonal” O∆ . In keeping with the following sections, rather than η, we prefer to work with its image u = I 2 (η) ∈ HT 2 (S) under the HKR isomorphism. Also, we assume that u = (γ, 0, 0), with V γ ∈ H 0 ( 2 T M ), which is the only somewhat subtle case. The theory of quasi-coherent sheaves on first-order noncommutative deformations mirrors that for schemes [To, §3,4]. In particular, over any affine U , such a sheaf M is determined by its sections M (U ), and its sections over a principal affine Uf are precisely the localization M (U )f [To, Lemma 3.1, Def. 4.1]. Fix an ample line bundle L on S; for any section f ∈ Γ(S, L n ), n ∈ Z, let Sf be the affine open where f does not vanish. Consider the affine open covering C ′ of S given by {Sf : f ∈ Γ(S, L n ), n ∈ Z}. Note that C ′ is closed under intersection, and that given any two elements of C ′ , their intersection is a principal affine in each of them. Consider the affine open covering C := {U × V : U, V ∈ C ′ } of S × S. Let D stand for the dual numbers over C. Denote the class −p∗1 (γ) + p∗2 γ by −γ ⊞ γ. Note −γ⊞γ that given U × V ∈ C , OS×S (U × V ) = OS−γ (U ) ⊗D OSγ (V ). For each U × V ∈ C , let AU ×V γ be the D-flat coherent OU−γ⊞γ ×V module whose sections over (U × V ) are OS (U ∩ V ). The right −γ γ γ OS (U )⊗D OS (V )-module structure of OS (U ∩ V ) is given by restriction to U × V , followed by left and right multiplication in the ring OSγ (U ∩ V ). We claim that the coherent sheaves AU ×V glue to give a coherent sheaf O∆ over S × S. It suffices to check that for any U × V ∈ C ′ , and principal affine subsets Ug ⊂ U , Vh ⊂ V , there is a canonical identification between the

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modules AU ×V (Ug × Vh ) and AUg ×Vh (Ug × Vh ). The former is the localization with respect to the left OSγ (U ∩ V ), right OSγ (U ∩ V ) bimodule structure, OSγ (U ∩ V )g⊗h . The latter is OSγ (U ∩ V )gh , the localization being with respect to the right OSγ (U ∩ V )-module structure. Given any two local sections a, b of OS ⊗C D, a ∗γ b = b ∗−γ a, so these two localizations are equal. 7.2. A map on tangent spaces. We wish to compare the Hochschild cohomologies of X and MH (v), but as the functor ΦU : D b (X) → D b (MH (v) is not an equivalence, we do not a priori have a homomorphism φ : HH ∗ (X) → HH ∗ (M ). Nevertheless, the natural construction (7.1) can be modified to give a workable map between the degree 2 components of these groups: Construction 7.8. Write V ∈ D b (M × X) for U ∨ [2], the kernel of the adjoint to ΦU . As in (7.1), given η ∈ HH 2 (X), one obtains a map: F = U ◦ O∆X ◦ V

U ◦η◦V

−→ U ◦ O∆X [2] ◦ V = F [2]. ǫ

Applying the functor HomM 2 (v) ( , O∆M [2]) to the triangle E [1] → F → O∆M , we obtain the sequence Hom(E , O∆M ) → Hom(O∆M , O∆M [2]) → Hom(F , O∆M [2]) → Hom(E , O∆M [1]).

The first and last groups are (0). Assuming this for the moment, set φHH (η) to be the unique lift of ǫ[2](U ◦η ◦V ) ∈ HomM 2 (v) (F , O∆M [2]) in Hom(O∆M , O∆M [2]). This defines the desired map φHH : HH 2 (X) → HH 2 (M ).

(7.3)

We prove the claim using the spectral sequence

E2p,q = Homp (H−q (∆∗M E ), OM ) =⇒ Homp+q (E , O∆M ) and Proposition 4.2. Observe that H −1 (∆∗ (E ), OM ) ∼ = T or1 (E , O∆ ) vanishes by the latter M

M

result. Hence, E20,1 vanishes. The question reduces to proving the vanishing of the terms E2i,0 := HomiM (Ω2M /OM · σ, OM )

for i = 0, 1. This follows from the facts that the sheaf (Ω2M /OM · σ) is a self-dual direct summand of Ω2M , and that H i (Ω2M /OM · σ) vanishes for i = 0, 1. We can say this slightly differently: There are natural maps (7.4)

HH 2 (X)

U◦

/ HomX×M (U , U [2]) o

◦U

HH 2 (M ),

the left map is an injection, while the right is an isomoprhism. The map φHH is the one obtained by composing the first with the inverse of the second. Hochschild homology is naturally a module over Hochschild cohomology, the action being composition. This structure gives rise to the homomorphisms mX mM

: HH 2 (X) → Hom(HH0 (X), HH−2 (X))

: HH 2 (M ) → Hom(HH0 (M ), HH−2 (M )).

We note that mX is an isomorphism and mM is injective. By functoriality of Hochschild homology, we also have the maps Φ∗ := ΦU∗ Ψ∗ := ΨU∗

: HH∗ (X) → HH∗ (M ), and : HH∗ (M ) → HH∗ (X).

The map φHH intertwines the Hochschild module structures via Φ∗ and Ψ∗ :

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E. MARKMAN AND S. MEHROTRA

Lemma 7.9. The following diagram is commutative for every λ ∈ HH 2 (X). HHi (M )

Ψ∗

/ HHi (X)

mM (φHH (λ))

Φ∗

/ HHi (M ) mM (φHH (λ))

mX (λ)



HHi−2 (M )

Ψ∗



/ HHi−2 (X)

Φ∗



/ HHi−2 (M )

Proof. Let λX be a class in HH 2 (X) and λM a class in HH 2 (M ) satisfying the equality U ◦ λX = λM ◦ U in HomX×M (U , U [2]). Then the following diagram commutes HHi (X)

Φ∗

/ HHi (M ) mM (λM )

mX (λX )





HHi−2 (X)

Φ∗

/ HHi−2 (M ),

by the proof of [AT, Prop. 6.1]. The above statement holds and its proof applies without the assumption that the right map in diagram (7.4) is an isomorphism. The statement thus applies also to the functor ΨU . Apply the statement with λX = λ and λM = φHH (λ) to verify the commutativity of both squares in the statement of the Lemma.  The endomorphism Φ∗ Ψ∗ of HH∗ (M ) is self-adjoint with respect to the Mukai pairing. It n−1 satisfies (Φ∗ Ψ∗ )2 = nΦ∗ Ψ∗ . Indeed, the kernel of ΨU ◦ ΦU is the direct sum ⊕i=0 O∆ [−2i] by Assumption 7.2, and ΨU (ΦU (x)) = x⊕x[−2]⊕· · ·⊕x[2−2n] for any object x in D b (X). Hence, Ψ∗ Φ∗ is multiplication by n. The subspace Im(Φ∗ ) is the eigenspace of Φ∗ Ψ∗ with eigenvalue n and the subspace ker(Ψ∗ ) is the eigenspace with eigenvalue 0. We get the orthogonal direct sum decomposition HH∗ (M ) = Im(Φ∗ ) ⊕ ker(Ψ∗ ). Let E be an H-stable sheaf on X with Mukai vector v ∈ HH0 (X). Let α ∈ HH0 (M ) be the Mukai vector of ΦU (E ∨ [2]) and let β ∈ HH0 (M ) be the Mukai vector of the sky-scraper sheaf of a point. Given a subset Σ of HH0 (M ), denote by ann(Σ) the subspace of HH 2 (M ) consisting of classes ξ, such that mM (ξ)(c) = 0, for all c ∈ Σ. We use the analogous notation for subsets of HH0 (X). Lemma 7.10. (1) The image of φHH is equal to the subspace of HH 2 (M ) consisting of classes ξ, such that mM (ξ) commutes with Φ∗ Ψ∗ . (2) The following equality of subspaces of HH 2 (M ) holds: (7.5)

φHH (ann(v ∨ )) = ann{α, β}.

(3) The normalized HKR isomorphism maps the subspace φHH (ann(v ∨ )) of HH 2 (M ) into the direct sum H 1 (T M ) ⊕ H 2 (OM ) in HT 2 (M ). The image is equal to the subspace

(7.6)

℧ := {(ξ, θ) : ξ ∈ H 1 (T M ), θ ∈ H 2 (OM ), and ξ · c1 (α) + (2 − 2n)θ = 0}.

Proof. (1) Φ∗ Ψ∗ commutes with mM (φHH (λ)), for all λ ∈ HH 2 (X), by Lemma 7.9. We know that φHH is injective and that its image has codimension 1. Hence it remains to exhibit classes ξ of HH 2 (M ), which do not commute with Φ∗ Ψ∗ . ΨU maps the sky-scraper sheaf of the point [E] corresponding to the isomorphism class of the sheaf E to the object E ∨ [2] in D b (X). Let e be the class of E ∨ [2] in HH0 (X). Then α = Φ∗ (e), Ψ∗ (α) = ne, and Ψ∗ (β) = e. Hence, Ψ∗ (α − nβ) = 0.

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65

The class c1 (α) does not vanish [Ma5, Lemma 7.2]. Hence, there exists a class ξ of H 1 (T M ), such that ξ · c1 (α) is a non-zero class in H 2 (M, OM ) ⊂ HΩ−2 (M ). Considering ξ as a class in HH 2 (M ), via the HKR isomorphism, then ξ belongs to ann(β), but it does not belong to ann(α). Hence, mM (ξ)(α − nβ) 6= 0. Assume that mM (ξ) commutes with Φ∗ Ψ∗ . Then Im(Φ∗ ) and ker(Ψ∗ ) are invariant with respect to mM (ξ). In addition, we have mM (ξ)(α − nβ) = mM (ξ)(α)

The left hand side belongs to ker(Ψ∗ ), since α − nβ does, and the right hand side belongs to the image of Φ∗ , since α does. Hence, mM (ξ)(α) vanishes and ξ belongs to ann(α). A contradiction. Hence, mM (ξ) does not commute with Φ∗ Ψ∗ . (2) Let λ ∈ HH 2 (X) be a class, such that φHH (λ) belongs to ann(β). Then φHH (λ) commutes with Φ∗ Ψ∗ and so φHH (λ) belongs also to ann(α), as shown in the proof of part (1). Hence, the intersection Im(φHH ) ∩ ann(β) is contained ann{α, β}. Note that both ann(β) and Im(φHH ) are hyperplanes in HH 2 (M ). We have seen above that ann(β) is not contained in ann(α). We conclude the equality (7.7)

Im(φHH ) ∩ ann(β) = ann{α, β}.

Furthermore, the hyperplanes Im(φHH ) and ann(β) are distinct and ann{α, β} has codimension 2 in HH 2 (M ). The equality φHH (ann(v ∨ )) = Im(φHH ) ∩ ann(α),

follows from the commutativity of the right square in Lemma 7.9 and the injectivity of Φ∗ . Hence, if ann(α) is a hyperplane in HH 2 (M ), then the hyperplanes ann(α) and Im(φHH ) are distinct. We conclude that either ann(α) is contained in ann(β), or ann(α) is a hyperplane and the three hyperplanes ann(α), ann(β), and Im(φHH ) are distinct. If H1 , H2 , H3 are three distinct hyperplanes in a vector space, such that H1 ∩ H2 and H2 ∩ H3 are equal to the same subspace W , then H1 ∩ H3 = W . The equality Im(φHH ) ∩ ann(α) = ann{α, β}

thus follows from (7.7). The above equality is clear if ann(α) is contained in ann(β). The equality (7.5) follows from the last two displayed equalities. (3) The HKR isomorphism maps ann(α) into the hyperplane in HT 2 (M ), consisting of classes (π, ξ, θ), π ∈ H 0 (∧2 T M ), ξ ∈ H 1 (T M ), θ ∈ H 2 (OM ), satisfying π · α2 + ξ · c1 (α) + (2 − 2n)θ = 0,

where α2 is the graded summand in H 2 (Ω2M ) of the image via the HKR isomorphism of the Mukai vector α, since the rank of α is 2 − 2n. Indeed the latter hyperplane is the kernel of the composition HT 2 (M ) → HH−2 (M ) → H 2 (OM ) of pairing with α followed by projection on the direct summand H 2 (OM ). (We are using here the proof of Calduraru’s conjecture about the isomorphism of harmonic and Hochschild strucures [CRVdB].) The HKR isomorphism maps ann(β) onto the direct sum H 1 (M, T M ) ⊕ H 2 (M, OM ). The statement now follows from part (2).  7.3. The comparison. Given u ∈ HT 2 (X), it will be convenient to adopt the simpler notation D b (X, u) for the deformed category D b (Coh(X, u)); a similar notation is used for the corresponding categories on M . Let φT : HT 2 (X) → HT 2 (M ) denote the conjugate 2 )−1 ◦ φHH ◦ I 2 . Also let ˇ : HT ∗ (X) → HT ∗ (X) be the operation which on a homoge(IM X neous t ∈ H p (∧q TX ) is defined as tˇ := (−1)q t. For u ∈ HT ∗ (X), and w ∈ HT 2 (M ), the class ∗ u + π ∗ w ∈ HT 2 (X × M ) will be denoted by u ⊞ w; the same notation will be followed for πX M the other Cartesian products, such as M × M , that appear.

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Lemma 7.11. Let u ∈ HT 2 (X), and w = φT (u) ∈ HT 2 (M ). There is a perfect object U 1 ∈ D b (X × M, −ˇ u ⊞ w) whose derived restriction i∗ (U 1 ) is isomorphic to U .

Proof. By Theorem 7.4, it suffices to show that the degree 2 piece of (−ˇ u ⊞ w) · exp a(U ) is 0. We repeat Toda’s calculation of this obstruction in our setting. According to [To, Lemma 5.8] (see Remark 7.12 below), the following diagrams commute: HT ∗ (X × M )

× exp a(U )

/ Hom∗ X×M (U , U ) O

O

∗ πX

U◦

HT ∗ (X) HT ∗ (X × M )

× exp a(U )

/ Hom∗ (U , U ) X×M O

O

∗ πM

HT ∗ (M )

/ HH ∗ (X)

∗ τ∗ IX

◦U ∗ IM

/ HH ∗ (M )

The symbol τ∗ in the first diagram is the involution of HH ∗ (X) arising from the interchange of ∗ (t) = I ∗ (tˇ) (see the discussion preceding Proposition the factors of X × X. One sees that τ∗ IX X 6.1 of [To]), so that ∗ ∗ ∗ ∗ (−πX u ˇ + πM w) · exp a(U ) = −U ◦ τ∗ IX (ˇ u) + IM (w) ◦ U

∗ ∗ = −U ◦ IX (u) + (φHH (IX (u))) ◦ U .

(7.8)

∗ (u)) ◦ U = U ◦ I ∗ (u). The result The description (7.4) of the map φHH yields that φHH (IX X follows immediately. 

Remark 7.12. Although it is assumed everywhere in [To] that the fuctor ΦU is an equivalence, Lemma 5.8 of that article holds without this requirement. Indeed, Toda only works with the + compositions exp a(U )X and exp a(U )M rather than the morphisms exp(a)+ X and exp(a)M (see p. 212, op. cit.), and while the latter morphisms may not exist without the assumption that ΦU is an equivalence, the former always do. −ˇ u,w be the deformed structure sheaf of the product X × MH (v), and write V 1 for Let OX×M −ˇ u,w RH om∗ (U 1 , OX×M )[2]. Note that derived dualization is defined on the full subcategory of perfect complexes of D b (X × M, −ˇ u ⊞ w), and sends it into the subcategory of perfect b complexes of D (M × X, −w ˇ ⊞ u): −ˇ u,w RH om∗ ( , OX×M ) : Dperf (X × M, −ˇ u ⊞ w) → Dperf (M × X, −w ˇ ⊞ u)

The functors corresponding to restriction and extension of scalars between various categories will be simply denoted i∗ and i∗ , without reference to the underlying spaces. ˇ Lemma A.5 of [MSM] implies that Consider the convolution U 1 ◦V 1 ∈ D b (M ×M, −w⊞w). ∗ ∼ ˇ ⊞ w ∈ HT 2 (M × M ), i (U 1 ◦ V 1 ) = F . We conclude that F deforms along the direction −w ′ w = φT (u), for any u ∈ HT 2 (X). Denote this infinitesimal deformation U 1 ◦ V 1 by F u . The object F is the restriction to a fiber of the family F constructed in §6.1 by Assumtion f0 such that F ∼ 7.2, that is, there exists a triple (M, η, A) ∈ U ⊂ M = F |M ×M . Given a class Λ 1 ξ ∈ H (T M ), let Mξ denote the first-order infinitesimal defomation of M in the direction of ξ. Let F ξ be the restriction of F to the fiber square M2ξ of Mξ over the length 2 subscheme f0 detrmined by ξ. of M Λ

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67

Lemma 7.13. (1) Let ξ ∈ H 1 (T M ). The infinitesimal deformation F ξ of F is an object of the derived category D b (M × M, −w ˇ ⊞ w), where w = (0, ξ, θ) is a class in the 2 subspace ℧ ⊂ HT (M ) defined in Lemma 7.10 (3). ′ (2) Write u for the class (φT )−1 (w) ∈ HT 2 (X). There is an isomorphism F u ∼ = F ξ. Proof. (1) Fix a point [E] ∈ M . The restriction F |M ×{[E]} is isomorphic to ΦU (E ∨ [2]). Making use of the fact proven in Lemma 4.1 that τ ∗ F ∼ = F ∨ [2], we see that the Chern character of F has the form: ch(F ) = (2 − 2n) + (−π1∗ (c1 (α)) + π2∗ (c1 (α))) + ch2 (F ) + · · · ∈ ⊕H i (ΩiM ×M ).

The product (−w ˇ ⊞ w) · ch(F ) in π1∗ (H 2 (OM )) ⊕ π2∗ (H 2 (OM )) ∼ = H 2 (OM ×M ) vanishes if and only if w ∈ ℧. This product is nothing but the trace of the obstruction class (−w ˇ ⊞ w) · exp a(F ) ∈ Hom(F , F [2]) [HuL, 10.1.6]. As F deforms, Theorem 7.4 implies that this class must indeed vanish, which completes the proof. (2) This should follow immediately from the infinitesimal rigidity of F from known results9. However, we were not able to locate a precise reference, so we sketch a short argument specific ′ to our case. Consider the object F u . Being a deformation of an object with cohomology sheaves in degrees contained in the interval [-1,0], it is perfect with cohomology in degrees ′ contained in [-1,0]. So, H0 (F u ) is perfect, hence flat over the dual numbers; using the ob′ servation of the previous sentence again, we get that H−1 (F u ) is flat over the dual numbers. Therefore, ′ ′ i∗ (H0 (F )) ∼ = (H0 (i∗ F )) ∼ = O∆ . u

u

M

′ i∗ (H−1 (F u ))

∼ Similarly, = E . The (twisted) sheaves O∆M and E are infinitesimally rigid ′ ′ (see Lemma 5.2). This implies that H0 (F u ) ∼ = O∆Mξ and H−1 (F u ) ∼ = E ξ , where E ξ is the ′

restriction of E to M2ξ [MM2, Lemma 4.11]. It follows that the object F u is an extension of O∆Mξ by E ξ [1]. Step 3 of Proposition 6.4 says that, up to scalars, there is a unique such extension. The claimed isomorphism follows from this.  Lemma 7.14. The functor ΨV 1 : D b (M, w) → D b (X, u) is right adjoint to the functor ΦU 1 : D b (X, u) → D b (MH (v), w). Moreover, the unit η : 1Db (X,u) → ΨV 1 ΦU 1 of this adjunction is split, that is, there exits a natural transformation ζ : ΨV 1 ΦU 1 → 1Db (X,u) such that ζη ∼ = 1Db (X,u) . Proof. The exact sequence 0 → C → C[ε]/(ε2 ) → C → 0 yields the triangle (7.9)

i∗ i∗ V 1 ◦ U 1 → V 1 ◦ U 1 → i∗ i∗ V 1 ◦ U 1

u ⊞ u) is the convolution of V 1 with U 1 . We have the where V 1 ◦ U 1 ∈ D b (X × X, −ˇ ∗ ∼ isomorphism i∗ i V 1 ◦ U 1 = i∗ V ◦ U by base-change [MSM, Lemma A.5]. u ⊞ u) for the structure sheaf of the diagonal. This is defined Write O∆X ∈ D b (X × X, −ˇ u , with the above when the gerby part θ ∈ H 2 (OX ) of u is 0. In general, set O∆X = ∆∗ OX −ˇ u⊞u OX×X -module structure defined in Example 7.7. This makes sense as a (−θ ⊞ θ)-twisted sheaf because the class (−θ ⊞ θ) restricts to the trivial class along the diagonal in S × S. 9One possible approach might be by combining [Lieb1, Theorem 3.1.1] and [Lieb2, Prop. 2.2.4.9]. The

former studies deformation theory for complexes, the latter for twisted sheaves; however, only the algebraic situation is considered.

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Consider the following sequence arising from applying the functor HomDb (X×X,−ˇu⊞u) (O∆X , to (7.9): 0 → HomX 2 (O∆X , V ◦ U ) → HomDb (X 2 ,−ˇu⊞u) (O∆X , V 1 ◦ U 1 ) → HomX 2 (O∆X , V ◦ U ) → 0 Ln−1 As V ◦ U ∼ = i=0 O∆X [−2i], we see that HomX 2 (O∆ , V ◦ U ) ∼ = HH 1 (X) = (0), = C, HomX 2 (O∆ , V ◦ U [1]) ∼ X

X

from which it follows that the sequence above is exact. Let η 1 : O∆X → V 1 ◦ U 1 be a lift of the unit η. We claim that η 1 is the unit of an adjunction ΦU 1 ⊣ ΨV 1 . Indeed, let A , B ∈ D b (X × X, −ˇ u ⊞ u), and denote their derived restrictions to D − (X × X) by A ,B, respectively. Consider the commuting diagram: Hom(ΦU 1 A , i∗ B)

/ Hom(Φ A , B) U1

/ Hom(Φ A , i∗ B) U1



 / Hom(Ψ Φ A , Ψ B) V1 U1 V1

 / Hom(Ψ Φ A , Ψ i∗ B) V1 U1 V1

Hom(ΨV 1 ΦU 1 A , ΨV 1 i∗ B) ◦η1

◦η1



Hom(A , ΨV 1 i∗ B)

◦η1





/ Hom(A , Ψ B) V1

/ Hom(A , Ψ i∗ B) V1

The first and third columns can be identified with the composition ◦η

HomX 2 (ΦU A , B) → HomX 2 (ΨV ΦU A , ΨV B) → HomX 2 (A , ΨV B)

by flat base-change and the adjunction i∗ ⊣ i∗ , which is clearly an isomorphism. It now follows by Proposition 1.1 of [Ha-RD] that the composition of the arrows in the central column is an isomorphism. As this isomorphism is bi-functorial in A and B, the claim is proved. Let H1 be the cone of the map η 1 : (7.10)

η

O∆X →1 V 1 ◦ U 1 → H1 .

As above, we can compute the extension group HomDb (X 2 ,−ˇu⊞u) (H1 , O∆X [1]) by applying the functor HomDb (X 2 ,−ˇu⊞u) (H1 , ) to the triangle i∗ O∆X → O∆X → i∗ O∆X and chasing the resulting long exact sequence. Using the fact that HomX 2 (H1 , i∗ O∆X [1]) ⊂ HH odd (X) = (0), the result is that this group is (0). In particular, triangle (7.10) is split, from which the second statement of the lemma follows immediately.  2 )−1 (ann(v ∨ )), and set w = φT (u) ∈ HT 2 (M ); the class w has the form (0, ξ, θ) Fix u ∈ (IX by Lemma 7.10. As above, let F ξ the restriction of F to M2ξ . Note that F ξ , together with the structure maps ǫ1 and δ1 , defines a comonad hL1 , ǫ1 , δ1 i on D b (M, w) by Theorem 1.8.

Theorem 7.15. (Theorem 1.9 (2)) There is an exact equivalence of triangulated categories between D b (X, u) and the category of comodules D b (M, w)L 1 . u ⊞ w constructed in Proof. Let U 1 be the deformation of U corresponding to the class −ˇ Lemma 7.11. Denote the comonad arising from the adjoint pair ΦU 1 ⊣ ΨV 1 by hL′1 , ǫ′1 , δ1′ i. The unit η : 1Db (X,u) → ΨV 1 ΦU 1 is split by the previous lemma. Therefore, the categories ′ D b (M, w)L 1 and D b (X, u) are equivalent by the Barr-Beck Theorem for triangulated categories [E, MS, Bal2], which says that for a split adjunction the comparison functor is an equivalence (see the discussion preceding the statement of Proposition 1.4). To prove the result, it only remains to show that there is an isomorphism of comonads between L1 and L′1 .

)

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69



We have seen above that the kernel F u of the functor L′1 is isomorphic to F ξ (Lemma 7.13); ′ ′ fix an isomorphism µ : F u f →F ξ . Note that Hom(F u , F ξ ) ∼ = Hom(F ξ , F ξ ) = C[ε]/(ε2 ) by Lemma 6.7. Let E be the universal twisted sheaf. Step 3 of the proof of of Proposition 6.4 shows that H omΠ (O∆M , E [1]) vanishes over U . Applying the functor RH omΠ ( , O∆ ) to the exact triangle E → F → O∆ ′ ∼ Hom(F ξ , O ) = ∼ C[ε]/(ε2 ). So we may modify allows one to conclude that Hom(F u , O∆1 ) = ∆1 µ by a scalar in order that the following diagram commutes: ′

Fu

ǫ′1

/O ∆1

µ





ǫ1

/O ∆1

It only remains to check that the isomorphism µ is compatible with the comultiplication maps δ1′ and δ1 , that is, the left square in the diagram below commutes. Use the equalities (ǫ′1 ◦ 1F ′ ) ◦ δ1′ = 1F ′ and (ǫ1 ◦ 1F ξ ) ◦ δ1 = 1F ξ , and the commutativity of the right square to u u conclude that the composition (ǫ1 ◦ 1F ξ ) ◦ (µ ◦ µ) ◦ δ1′ = µ. ′ Fu

δ1′

µ





/ F′ ◦ F′ u u

ǫ′1 ◦1F ′

u

/ F′ u µ

µ◦µ δ1



/ Fξ ◦ Fξ

ǫ1 ◦1F

ξ



/ Fξ

Thus, (µ ◦ µ) ◦ δ1′ ◦ µ−1 is a section of ǫ1 ◦ 1F ξ . The section δ1 is unique as composition by

→Hom(F ξ , F ξ ) by Assumption 7.2 and ǫ1 ◦ 1F ξ gives an isomorphism Hom(F ξ , F ξ ◦ F ξ )f Lemma 6.7. We conclude the equality δ1 = (µ ◦ µ) ◦ δ1′ ◦ µ−1 , proving the commutativity of the left square.  8. Variations of Hodge structures

Let X be a K3 surface, M := MH (v) a moduli space of sheaves on X, A the modular Azumaya algebra over M × M , and assume that (M, η, A) belongs to the open set U of f0 in Theorem 1.8, for some marking η (see Remark 6.17). The Hodge structure of the M Λ Mukai lattice of X can be deformed, as we deform the category D b (M, θ)L , of comodules for the comonad (L, ǫ, δ), along deformations of (M, L, ǫ, δ). These deformations of the Hodge structure are defined in [Ma4, Theorem 1.10]. The deformed Mukai lattice should be related to the Hochschild homology of the category D b (M, θ)L , and the type (1, 1) sublattice should be related to the numerical lattice of the K-group of D b (M, θ)L . The Mukai lattice of MH (v), as defined in [Ma4, Theorem 1.10], is that of X, by [Ma4, Theorem 1.14]. As we deform the Mukai lattice of MH (v), the class v remains of Hodge-type (1, 1), while its orthogonal complement remains isometric to H 2 (M, Z), by [Ma4, Theorem 1.10]. In particular, the class v spans the sublattice of integral classes of Hodge type (1, 1), for the Mukai lattice of a generic deformation of M . This agrees with Theorem 1.9 in the current paper, which suggests that the family of deformation we get is the complete family of deformations of D b (X), which preserve the Hodge type of the class v. Such deformations include deformations of D b (X),

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Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003 E-mail address: [email protected] ´ ticas, PUC Chile, Av. Vicun ˜ a Mackenna 4860, Santiago, Chile; Chennai Facultad de Matema Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, India E-mail address: [email protected]